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Measurement & SI units

From NOTES Intro. Physics is built on measurement, and every measurement is a number plus a unit. Drop the unit and the answer is incomplete.
  • The SI system has seven base units; the three used most in mechanics are the meter (m) for length, the kilogram (kg) for mass, and the second (s) for time.
  • The other base units are the ampere (A), kelvin (K), mole (mol), and candela (cd).
  • Derived units are combinations of base units: speed is m/s, and force is the newton, $N = kg·m/s^2$.
  • Quantities that are added or subtracted must have the same units, and a final answer must carry the correct unit.
  • Volume: $1 cm^3 = 1 mL$ and $1 m^3 = 1000 L$; for water, $1 mL = 1 g$.

SI prefixes & conversions

From NOTES Intro. Prefixes scale a unit by powers of 10, which is what makes the metric system a decimal system.
  • tera (T) = $10^{12}$, giga (G) = $10^9$, mega (M) = $10^6$, kilo (k) = $10^3$.
  • centi (c) = $10^{-2}$, milli (m) = $10^{-3}$, micro (μ) = $10^{-6}$, nano (n) = $10^{-9}$, pico (p) = $10^{-12}$.
  • To convert, replace the prefix with its power of 10: $3.5 km = 3.5 × 10^3 m$ and $1 MW = 10^6 W$.
  • Handy sizes: an adult is about 1.75 m tall, a classroom is about 10 m long, a physics textbook has a mass near 1 kg, and a high school student is about 60 kg.

Scientific notation & order of magnitude

From NOTES Intro. Scientific notation keeps very large and very small numbers manageable and makes calculations mechanical.
  • Write a number as $M × 10^n$ with $1 ≤ M < 10$ and n an integer.
  • Move the decimal point until one nonzero digit is left of it; the number of places moved is the exponent (positive for big numbers, negative for small ones).
  • To add or subtract, first rewrite both terms with the same power of 10, then combine the M values: $4.1×10^{-6} − 0.3×10^{-6} = 3.8×10^{-6}$.
  • To multiply, multiply the M values and add the exponents; to divide, divide the M values and subtract the exponents. Multiply and divide the units too.
  • Order of magnitude is the power of 10 closest to the value: $1284 kg = 1.284×10^3$ kg is order $10^3$, but $8756 kg = 8.756×10^3$ kg is closer to $10^4$.
  • Estimate with benchmarks: one building story ≈ 3 m, so a 30-story building ≈ 90 m ≈ $10^2$ m; a football field ≈ 100 m = 1000 dm.
  • Small benchmarks: a little finger is about 1 cm = 0.01 m wide; a pencil is about 5 g = $5.0 × 10^{-3}$ kg; an adult is about 70 kg.
  • Watch: Scientific Notation & Orders of Magnitude · Estimates and Order of Magnitude Calculations (YouTube)

Scalars vs. vectors

From NOTES Vectors and Scalars. Some quantities need only a size; others also need a direction.
  • A scalar has magnitude only: mass, length, time, distance, speed, density, energy, temperature.
  • A vector has magnitude and direction: displacement, velocity, acceleration, force, momentum, electric and magnetic field strength.
  • A vector is drawn as an arrow: its length shows the magnitude (never negative) and the tail-to-head direction shows the direction.
  • In print a vector is bold; handwritten, an arrow goes over the letter.
  • Concurrent vectors act on the same point at the same time.
  • Watch: Introduction to Tip-to-Tail Vector Addition, Vectors and Scalars (YouTube)

Adding & subtracting vectors graphically

From NOTES Vectors and Scalars. The sum of vectors is called the resultant, and it can be found by drawing.
  • Head-to-tail: place the second vector's tail at the first one's head; the resultant runs from the first tail to the last head.
  • Parallelogram: draw both vectors from a common tail and complete the parallelogram; the diagonal from that tail is the resultant.
  • Order does not matter: $A + B = B + A$. In one dimension, just add with signs, so opposite directions partly cancel.
  • To subtract, add the negative: $A − B = A + (−B)$, where $−B$ has the same length but points the opposite way.
  • Multiplying by a positive scalar changes only the length; a negative scalar also reverses the direction.
  • The resultant of two vectors is largest ($A + B$) at 0° between them, smallest ($|A − B|$) at 180°, and shrinks steadily as the angle grows from 0° to 180°.
  • Watch: Adding and Subtracting Vectors Made Simple (YouTube)

Perpendicular vectors & the Pythagorean theorem

From NOTES Vectors and Scalars. When two vectors are at right angles, the resultant comes straight from a right triangle.
  • Magnitude: $R = \sqrt{A^2 + B^2}$.
  • Direction: $\tanθ = B/A$, where θ is measured from vector A toward B.
  • Example: 5 m east then 2 m north gives $R = \sqrt{29} ≈ 5.4$ m at about 22° north of east.
  • Boat aimed straight across a river: 10 km/h in still water with a 4 km/h current gives $\sqrt{10^2 + 4^2} ≈ 10.8$ km/h, angled downstream.
  • The equilibrant has the same magnitude as the resultant but points the opposite way; adding it makes the net vector zero.
  • Watch: How To Find The Resultant of Two Vectors (YouTube)

Components of a vector

From NOTES Vectors and Scalars. Any vector can be replaced by two perpendicular components, which turns angled problems into simple x and y arithmetic.
  • With θ measured from the horizontal: $A_x = A\cosθ$ and $A_y = A\sinθ$.
  • Example: 60. N at 30° above horizontal gives $A_x = 60\cos30° ≈ 52$ N and $A_y = 60\sin30° = 30$ N.
  • To add vectors analytically: resolve each into x and y components, add all the x parts and all the y parts separately, then combine them with the Pythagorean theorem.
  • As θ from the horizontal grows toward 90°, the horizontal component shrinks and the vertical one grows.
  • On an incline of angle θ, the weight's component parallel to the slope is $F_g\sinθ$.
  • Watch: How To Find The Components of a Vector Given Magnitude and Direction · Using Cosine and Sine to Find Vector Components (YouTube)

Dot & cross products (optional)

From NOTES Vectors and Scalars. Vectors can also be multiplied by each other in two ways, which matter later for work and magnetism.
  • Dot product: $A·B = AB\cosθ$ is a scalar, used for work and power; it is zero for perpendicular vectors.
  • Cross product: $A×B$ is a vector with magnitude $AB\sinθ$, perpendicular to both, found with the right-hand rule; it is zero for parallel vectors.
  • Order matters for the cross product: $B×A = −(A×B)$.

Measurement, precision & error

From AMSCO book. No measurement is exact, so physicists track how precise and how accurate it is.
  • Precision is the smallest decimal place in a measurement; the instrument's scale markings set it, and one estimated digit is read beyond the smallest marked interval.
  • Accuracy is closeness to the accepted value. Absolute error = |measured − accepted|; percent error = |measured − accepted| / accepted × 100%.
  • Systematic errors push every reading the same way (fix by calibrating); random errors scatter readings high and low (reduce by averaging many trials).
  • Significant figures: nonzero digits, zeros between digits, and trailing zeros after a decimal point count; leading zeros do not.
  • A calculated answer can be no more precise than the least precise measurement used.

Distance & speed

From NOTES Distance and Displacement – Speed and Velocity. These are the scalar ways to describe motion: how far, and how fast.
  • Distance is the total length of the path traveled; it is a scalar, never negative, measured in meters.
  • Average speed is total distance divided by total time: $v = d/t$, in m/s. Include any time spent stopped.
  • Instantaneous speed is how fast you are moving at one moment, like a speedometer reading.
  • Speed has no direction, so a car going around a circular track at 20 m/s has constant speed even though its direction keeps changing.
  • Watch: Distance, Displacement, Average Speed, Average Velocity (YouTube)

Displacement & velocity

From NOTES Distance and Displacement – Speed and Velocity. These are the vector versions: they track where you ended up relative to where you started.
  • Displacement is the straight-line distance from start to finish plus the direction, for example 5 m south.
  • In one dimension displacement is positive or negative depending on direction; it can be zero after a round trip, and its magnitude never exceeds the distance traveled.
  • Average velocity is displacement divided by time: $\bar v = Δd/Δt$, in m/s, in the direction of the displacement.
  • Instantaneous velocity is speed and direction at one instant (a speedometer plus a compass).
  • By convention forward, right, and up are positive; backward, left, and down are negative.
  • Constant velocity requires both constant speed and constant direction.
  • Watch: Speed vs. Velocity, Distance vs. Displacement (YouTube)

Graphing uniform motion

From NOTES Graphing constant motion. Uniform motion means constant velocity: the same speed in the same direction for the whole time interval.
  • On a velocity–time graph, constant velocity is a horizontal straight line (parallel to the time axis).
  • On a distance–time (or position–time) graph, constant velocity is a straight sloped line.
  • The slope of a d–t graph is the velocity: its size is the speed and its sign is the direction (positive slope = positive direction, negative slope = negative direction).
  • A horizontal line on a d–t graph means the object is not moving.
  • The area under a v–t graph is the distance traveled: for constant velocity, $d = vt$ is just the rectangle's area.
  • Watch: Position-Time Graphs: Calculating Slope (YouTube)

Math basics for physics

From class schedule (photos). Test #1 starts with the math every later problem uses: rearranging formulas, right-triangle trig, and unit conversions.
  • Solve for the unknown with symbols first, then substitute numbers with their units: $d = \tfrac12at^2$ becomes $t = \sqrt{2d/a}$.
  • Right-triangle trig (SOH-CAH-TOA): $\sinθ = opp/hyp$, $\cosθ = adj/hyp$, $\tanθ = opp/adj$; the Pythagorean theorem gives $c = \sqrt{a^2 + b^2}$.
  • Slope of a straight-line graph = rise ÷ run, with units (m ÷ s = m/s).
  • Convert units by multiplying by fractions equal to 1: $72 km/h × (1000 m / 1 km) × (1 h / 3600 s) = 20 m/s$ (divide km/h by 3.6).
  • Keep units through every step; if the final unit is wrong, the setup is wrong.
  • Watch: Trigonometry Made Easy: Cosine and Sine for Vector Components (YouTube)

Lab skills: can capacity & distance vs. displacement

From Labs 1.1 and 1.2 (photos). Labs 1.1 and 1.2 practice measuring carefully and telling a path from a straight-line change in position.
  • Lab 1.1: a can is a cylinder, so its capacity is $V = πr^2h$. Measure the diameter and halve it for r; using the diameter by mistake makes V four times too big.
  • $1 cm^3 = 1 mL$, so a can 6.6 cm across and 12.2 cm tall holds about $π(3.3)^2(12.2) ≈ 417 mL$.
  • Record every measurement with its unit and the instrument's precision, and compare with the accepted value using percent error.
  • Lab 1.2: walking a path, distance is the total length walked; displacement is the straight line from start to finish plus its direction.
  • Halfway around a 30 m × 40 m rectangle (to the opposite corner), the distance is 70 m but the displacement is 50 m.

Vector scale drawings & relative velocity

From vector handout (photos). Tip-to-tail diagrams are drawn to scale, so the length of each arrow stands for its magnitude.
  • Choose a scale and keep it for every vector: if 15 mm represents 30 m/s, then 1 mm = 2 m/s and 20 m/s is drawn 10 mm long.
  • Two vectors A and B combine to anything from $|A − B|$ (opposite directions, 180°) to $A + B$ (same direction, 0°): 3 m and 4 m give 1 m to 7 m, and 5 m at 90°.
  • As the angle between two vectors grows from 0° to 180°, their resultant shrinks steadily from the sum to the difference.
  • Relative velocity: add velocities as vectors. Walking forward at 4.0 m/s in a bus moving at 2.0 m/s gives 6.0 m/s relative to the street; walking toward the back gives 2.0 m/s the other way.
  • Measure the resultant's length and angle on the drawing, then convert back with the scale.

Acceleration

From NOTES Motion with acceleration. Acceleration is the rate of change of velocity. It is a vector, measured in $m/s^2$.
  • Average acceleration: $a = Δv/Δt = (v_f − v_i)/t$.
  • Velocity can change in speed, in direction, or both, so a car rounding a curve at constant speed is still accelerating.
  • Negative acceleration just points in the negative direction. Deceleration means the acceleration is opposite to the velocity, so the object slows down.
  • An object with negative velocity and negative acceleration is speeding up (in the negative direction).
  • Zero acceleration means constant velocity; the object may be at rest or moving.
  • For constant acceleration only: $v_f = v_0 + at$, $d = v_0t + \tfrac12at^2$, $v_f^2 = v_0^2 + 2ad$, and $d = \tfrac12(v_0 + v_f)t$.
  • Watch: Average Acceleration and Instantaneous Acceleration (YouTube)

Graphing motion with acceleration

From NOTES Graphing motion with acceleration. When velocity changes, the position–time graph curves, and the velocity–time graph carries the acceleration.
  • Nonuniform motion shows up as a curved line on a d–t graph.
  • Instantaneous velocity is the slope of the tangent line at one point on the d–t curve; average velocity is the slope of the straight line joining the start and end points.
  • Where the tangent is horizontal (slope zero) the object is momentarily at rest; where the slope changes sign, it has turned around.
  • The slope of a v–t graph is the acceleration: positive slope means the velocity is increasing, negative slope means it is decreasing, and a flat line means zero acceleration.
  • In the class notes: a positive slope is an increase in velocity in the positive direction (acceleration), and a negative slope is a decrease in velocity (the notes call this deceleration). On a straight v–t line the slope equals $\tan\alpha = a$, where $\alpha$ is the angle the line makes with the time axis. A horizontal line has slope zero, meaning no acceleration. Careful: an object truly slows down only when its acceleration points opposite to its velocity.
  • The area under a v–t graph is still the distance traveled, even when the line slopes (use triangles and rectangles).
  • $v = v_0 + at$ has the same form as $y = mx + b$: on a v–t graph the slope is $a$ and the vertical intercept is $v_0$.
  • Watch: Velocity Time Graphs, Acceleration & Position Time Graphs (YouTube)
Graph
Positive accelerationv (m/s)t (s) α v₀vt tan α = a > 0

Positive acceleration: the v–t line slopes upward, so velocity is increasing. Its slope is tan α = a > 0.

Graph
Negative accelerationv (m/s)t (s) α v₀vt tan α = a < 0

Negative acceleration: the v–t line slopes downward, so velocity is decreasing. Its slope is tan α = a < 0.

Graph
Zero accelerationv (m/s)t (s) v slope = 0, so a = 0 (constant velocity)

Zero acceleration: a horizontal v–t line has slope zero, so the velocity stays constant.

Graph
Position–time curve (speeding up)d (m)t (s)tangent = instantaneous vsecant = average v

On a curved d–t graph, the slope of the tangent at one point is the instantaneous velocity, and the slope of the straight line joining two points (the secant) is the average velocity between them.

Writing a numeric answer with no unit, or the wrong unit; a correct number can still lose half the credit.
Adding vector magnitudes directly (3 N + 4 N = 7 N) when the vectors are not in the same direction.
Using cos for the vertical component or sin for the horizontal when θ is measured from the horizontal.
Treating distance and displacement as the same thing; a round trip has zero displacement but nonzero distance.
Adding numbers in scientific notation without first matching the powers of 10.
Calling every negative acceleration 'slowing down': an object only slows when its acceleration points opposite to its velocity.
Averaging two speeds for a round trip (60 and 75 km/h → 67.5): average speed is always total distance ÷ total time (300 km ÷ 4.5 h ≈ 67 km/h).
Using the diameter instead of the radius in $V = πr^2h$.
SOH-CAH-TOA: $\sinθ = opp/hyp$, $\cosθ = adj/hyp$, $\tanθ = opp/adj$; km/h ÷ 3.6 = m/s.
$M × 10^n$ with $1 ≤ M < 10$; multiply → add exponents, divide → subtract exponents.
Percent error $= |measured − accepted| / accepted × 100\%$.
$R = \sqrt{A^2 + B^2}$ for perpendicular vectors; $|A − B| ≤ R ≤ A + B$ in general.
$A_x = A\cosθ$, $A_y = A\sinθ$ (θ from the horizontal).
Average speed $v = d/t$ (scalar); average velocity $\bar v = Δd/Δt$ (vector).
$a = Δv/Δt$; constant a: $v_f = v_0 + at$, $d = v_0t + \tfrac12at^2$, $v_f^2 = v_0^2 + 2ad$.
Slope of d–t = velocity; slope of v–t = acceleration; area under v–t = distance.
Cylinder (can): $V = πr^2h$; $1 cm^3 = 1 mL$. Resultant of A and B: $|A − B| ≤ R ≤ A + B$.
Diagram
θ A Ax = A cosθ Ay = A sinθ xy

A vector A at angle θ above the horizontal is the sum of its perpendicular components: A_x = A cosθ along the x-axis and A_y = A sinθ along the y-axis.

Distance vs. displacement

Kinematics begins by separating how far you traveled from how far you ended up from the start.
  • Distance is total path length; it is a scalar and never negative.
  • Displacement is the straight-line change in position, $Δx = x_f − x_i$; a vector with a sign in 1D.
  • A round trip has zero displacement but nonzero distance.
  • Choose a positive direction first; every sign in the problem is relative to that choice.
  • SI unit for both is the meter.

Speed vs. velocity

The rate words split the same way: one keeps direction, one does not.
  • Average speed = total distance / total time (scalar, ≥ 0).
  • Average velocity = displacement / time, $v_{avg} = Δx/Δt$ (vector).
  • Instantaneous velocity is the slope of the x–t graph at that instant.
  • Constant velocity means unchanging speed AND direction.
  • Unit: m/s.

Acceleration and motion graphs

Acceleration measures how quickly velocity changes; graphs make the relationships visible.
  • $a_{avg} = Δv/Δt$, units $m/s^2$.
  • Positive acceleration means velocity is becoming more positive — not necessarily speeding up.
  • Speeding up: a and v point the same way. Slowing down: opposite ways.
  • Slope of x–t is velocity; slope of v–t is acceleration; area under v–t is displacement.
  • A straight line on a v–t graph means constant acceleration.

The kinematic equations

Four equations link displacement, velocity, acceleration, and time when acceleration is constant.
  • $v = v_0 + at$
  • $Δx = v_0 t + 0.5 a t^2$
  • $v^2 = v_0^2 + 2 a Δx$
  • $Δx = 0.5 (v_0 + v) t$
  • Pick the equation missing the quantity you neither know nor want.

Free fall

Near Earth's surface, gravity alone gives every object the same downward acceleration.
  • g ≈ $9.8 m/s^2$ downward (air resistance neglected).
  • At the top of a vertical toss, velocity is zero but acceleration is still g downward.
  • Time up equals time down for a symmetric path; return speed equals launch speed.
  • Substitute a = −g (with up positive) into the kinematic equations.
  • A dropped object and one thrown horizontally from the same height fall for the same time.
  • Aristotle taught that heavier objects fall faster. Galileo showed that, without air resistance, all objects fall with the same acceleration.
  • In a vacuum tube a coin and a feather land together; on the Moon, Apollo 15's hammer and feather hit the ground at the same time.
  • Near Earth's surface $g ≈ 9.8 m/s^2$, always directed downward; with up as positive, use $a = −g$.
  • Dropped from rest: $v_f = gt$, $d = \tfrac12gt^2$, $v_f^2 = 2gd$.
  • Thrown straight up: the velocity decreases by 9.8 m/s every second, is zero at the top, and the acceleration is still $g$ downward the whole time.
  • Reaction time from a dropped ruler: $t = \sqrt{2d/g}$, where d is how far it fell before you caught it (about 0.2 s for most people).

Projectile motion

Two-dimensional launches are solved by treating horizontal and vertical motion as independent, sharing only time.
  • Horizontal: $a_x = 0$, so $v_x = v_0 \cosθ$ stays constant.
  • Vertical: $a_y = −g$, so the y-motion is free fall with $v_{0y} = v_0 \sinθ$.
  • Time to the peak = $v_{0y}/g$; total flight time on level ground = twice that.
  • Range on level ground $R = v_0^2 \sin(2θ)/g$, maximized at 45°; 30° and 60° tie.
  • Impact speed = $\sqrt{v_x^2 + v_y^2}$.
Using the kinematic equations when acceleration is not constant.
Thinking a projectile at its peak has a = 0 — it is still g downward.
Using the full launch speed as the horizontal speed instead of $v_0 \cosθ$.
Switching sign conventions partway through a problem.
Saying the acceleration is zero at the top of a vertical toss; the velocity is zero there, but the acceleration is still g downward.
Assuming heavier objects fall faster; without air resistance every object falls with the same acceleration.
Same launch height → same fall time, whatever the horizontal speed.
If time is the missing quantity, use $v^2 = v_0^2 + 2aΔx$.
Slope of x–t = velocity; slope of v–t = acceleration; area under v–t = displacement.
Split the projectile into x and y, solve each, reunite through the shared time t.
Free fall from rest: $v = gt$, $d = \tfrac12gt^2$, $t = \sqrt{2d/g}$, $g ≈ 9.8 m/s^2$.

Force and free-body diagrams

A force is a push or pull; a free-body diagram isolates one object and shows every force on it.
  • Force is a vector in newtons $(1 N = 1 kg·m/s^2)$.
  • Draw the object as a dot; draw each force as an arrow from the dot.
  • Common forces: weight (down), normal (perpendicular to surface), tension, friction, applied.
  • Include only forces acting ON the object, not forces it exerts on others.
  • Choose axes along and perpendicular to the acceleration.

Newton's first law (inertia)

An object keeps doing what it is doing unless a net force acts.
  • Zero net force means zero acceleration — at rest or constant velocity (equilibrium).
  • Inertia is resistance to a change in motion; mass measures it.
  • There is no 'force of motion' keeping a sliding object going; it coasts until friction stops it.
  • Equilibrium: $ΣF_x = 0$ and $ΣF_y = 0$.
  • A frame moving at constant velocity is also an inertial frame.

Newton's second law

Net force sets acceleration, scaled by mass.
  • $ΣF =$ ma, applied per axis: $ΣF_x = ma_x, ΣF_y = ma_y$.
  • Acceleration points in the direction of the net force.
  • Double the net force → double the acceleration; double the mass → half the acceleration.
  • Weight W = mg is a force, distinct from mass.
  • Keep units consistent: kg, $m/s^2, N$.

Newton's third law

Forces come in equal, opposite pairs acting on two different objects.
  • If A pushes B, B pushes A equally hard in the opposite direction.
  • The two forces in a pair never act on the same object, so they never cancel on one free-body diagram.
  • Action–reaction pairs are the same type of force.
  • Rocket propulsion: engine pushes gas back, gas pushes rocket forward.
  • The forces are equal even when the two masses differ greatly.

Friction and inclines

Friction opposes sliding; inclined planes require resolving weight into components.
  • Kinetic friction: $f_k = μ_k N$, opposing motion, roughly constant.
  • Static friction: $f_s$ ≤ $μ_s N$ — it adjusts up to a maximum, then the object breaks free.
  • $μ_s$ is usually greater than $μ_k$, so starting to slide takes more force than continuing.
  • On an incline of angle $θ$: N = mg $\cosθ$, and the gravity component along the incline is mg $\sinθ$.
  • Friction depends on the normal force and surfaces, not on contact area.
Treating action–reaction pairs as forces that cancel — they act on different objects.
Setting $f_s = μ_s N$ when the object is not on the verge of slipping (that is only the maximum).
Assuming the normal force always equals mg — it changes on inclines and in elevators.
Forgetting to resolve weight into components on an inclined plane.
$a = ΣF / m$ — net force first, then divide by mass.
Normal force is whatever it must be to prevent sinking into the surface; solve for it.
Static friction is a range (≤); kinetic friction is a value (=).
On an incline: $\sinθ$ pulls it down the slope, $\cosθ$ presses it into the slope.
Diagram
mg FN mg sinθ θ

Free-body diagram of a block on a frictionless incline: weight (mg) resolves into a component along the slope (mg sinθ) and a component into the surface (mg cosθ), balanced by the normal force.

Uniform circular motion

Moving in a circle at constant speed is still accelerating, because the direction changes.
  • Velocity is tangent to the circle and constantly changes direction.
  • Centripetal acceleration points to the center: $a_c = v^2 / r$.
  • Period T is the time for one revolution; $v = 2πr / T$.
  • Constant speed means no tangential acceleration — only centripetal.
  • Also $a_c = 4π^2 r / T^2$.

Centripetal force

Some real force must supply the inward pull that bends the path into a circle.
  • $ΣF$ toward the center = $m v^2 / r$.
  • 'Centripetal force' is a role played by tension, gravity, friction, or the normal force.
  • Car on a flat curve: static friction is the centripetal force.
  • Vertical circle at the top: mg $+ T = m v^2 / r$.
  • There is no outward 'centrifugal force' in a ground frame — that feeling is inertia.

Newton's law of universal gravitation

Every mass attracts every other mass along the line joining them.
  • $F = G m_1 m_2 / r^2$, with $G = 6.67 × 10^-11 N·m^2/kg^2$.
  • r is the center-to-center distance, not the surface separation.
  • Inverse-square: triple the distance → one-ninth the force.
  • The force on each mass is equal in magnitude, even if the masses differ.
  • Surface gravity: $g = G M / R^2$.

Orbits and satellites

A satellite is in continuous free fall; gravity is its centripetal force.
  • $G M m / r^2 = m v^2 / r$, so orbital speed v = sqrt(G M / r).
  • Larger orbits are slower.
  • Astronauts feel weightless because they fall with the station, not because gravity is absent.
  • Kepler's third law: $T^2$ is proportional to $r^3$ for one central body.
  • A geostationary orbit has a 24-hour period and one specific radius.

Gravitational field and weight

The field g gives the gravitational force per kilogram at a point in space.
  • g = F/m, units N/kg (numerically equal to $m/s^2$).
  • Weight = mg depends on local g, so it differs on the Moon or at altitude.
  • Mass is invariant; weight is not.
  • Field lines point toward the mass; closer spacing means a stronger field.
  • g decreases with altitude as $1/r^2$ from Earth's center.
Adding an outward centrifugal force to a free-body diagram in a ground frame.
Using surface-to-surface distance instead of center-to-center in $F = G m_1 m_2 / r^2$.
Claiming there is no gravity in orbit — it is nearly as strong as at the surface.
Forgetting that at the top of a vertical circle gravity helps supply the centripetal force.
Centripetal acceleration always points to the center; centrifugal force is not real here.
Inverse-square: distance ×3 → force ÷9.
An orbit is falling and missing the ground; farther orbits are slower.
g has two identical faces: $9.8 m/s^2$ and 9.8 N/kg.
Diagram
Fc (toward center) v (tangent)

Centripetal force always points toward the center of the circular path, perpendicular to the object's velocity, which is tangent to the circle.

Angular quantities

Rotation has its own kinematics that mirror straight-line motion.
  • Angular displacement $θ$ (radians), angular velocity $ω = Δθ/Δt (rad/s)$, angular acceleration $α (rad/s^2)$.
  • Link to linear motion: $s = r θ, v = r ω, a_t = r α$.
  • The rotational kinematic equations look exactly like the linear ones with $x→θ, v→ω, a→α$.
  • One revolution = $2π$ radians = 360°.
  • All points on a rigid rotating body share $ω$ and $α$ but have different linear speeds.

Torque

Torque is the rotational analog of force — the tendency of a force to cause rotation.
  • τ = $r F \sinθ =$ (lever arm) × (force), units N·m.
  • The lever arm is the perpendicular distance from the axis to the line of the force.
  • A force directed through the axis produces zero torque.
  • Torque has a sign — counterclockwise is conventionally positive.
  • More distance from the pivot gives more torque for the same force.

Rotational equilibrium

A balanced object has no net force and no net torque.
  • $Στ = 0$ and $ΣF = 0$ for static equilibrium.
  • You may take torques about any point — pick one that removes an unknown force.
  • Balance beam or seesaw: $m_1 g d_1 = m_2 g d_2$.
  • The weight of a uniform beam acts at its center.
  • Two equilibrium conditions let you solve for two unknown support forces.

Moment of inertia

Moment of inertia is rotational mass — resistance to a change in spin.
  • I depends on mass AND how far that mass sits from the axis (I is proportional to $r^2$).
  • Point mass: $I = m r^2$. Hoop: $I = M R^2$. Solid disk: $I = 0.5 M R^2$. Solid sphere: $I = 0.4 M R^2$.
  • Rotational Newton's second law: $Στ = I α$.
  • Mass far from the axis contributes much more than mass near it.
  • A figure skater pulling arms in lowers I.

Angular momentum and rotational energy

Spinning objects store kinetic energy and carry angular momentum that is conserved.
  • Rotational KE = $0.5 I ω^2$; a rolling object has $0.5 m v^2 + 0.5 I ω^2$.
  • Angular momentum $L = I ω$, units $kg·m^2/s$.
  • With no external torque, L is conserved: $I_1 ω_1 = I_2 ω_2$.
  • A skater speeds up when arms are pulled in because I drops and L is fixed.
  • A rolling object accelerates down a ramp slower than a frictionless sliding one.
Using the full distance to the force instead of the perpendicular lever arm in τ = $r F \sinθ$.
Treating moment of inertia as if it depended only on mass, not its distribution.
Forgetting rotational kinetic energy when analyzing a rolling (not sliding) object.
Assuming every point on a rotating disk has the same linear speed.
Torque = force × perpendicular distance to the pivot; through the pivot means no torque.
$Στ = I α$ is just F = ma in rotational form.
Pull mass inward → I down → $ω$ up (angular momentum conserved).
Rolling race: smaller $I/(MR^2)$ wins — sphere beats disk beats hoop.

Work

Work is energy transferred by a force acting through a displacement.
  • $W = F d \cosθ$, where $θ$ is the angle between force and displacement.
  • A force perpendicular to the motion (normal, centripetal) does zero work.
  • Work is a scalar in joules (1 J = 1 N·m).
  • Negative work (friction, $θ$ > 90°) removes energy from the object.
  • On a force-vs-position graph, work is the area under the curve.

Kinetic energy and the work–energy theorem

The net work done on an object equals its change in kinetic energy.
  • KE = $0.5 m v^2$, always ≥ 0, scalar, in joules.
  • $W_{net} = ΔKE = 0.5 m v_f^2 − 0.5 m v_i^2$.
  • Doubling speed quadruples kinetic energy.
  • Positive net work speeds an object up; negative net work slows it down.
  • Often faster than kinematics for finding a final speed.

Potential energy and conservation

Stored energy plus kinetic energy is constant when only conservative forces act.
  • Gravitational PE near Earth: PE = mgh, from a chosen reference height.
  • Elastic PE in an ideal spring: PE = $0.5 k x^2$.
  • $KE_i + PE_i = KE_f + PE_f$ with no friction or drag.
  • With friction: $KE_i + PE_i = KE_f + PE_f +$ energy lost to heat.
  • Choose two points and write the energy equation — the path between them does not matter.

Power

Power is the rate of doing work or transferring energy.
  • $P = W/t = ΔE/Δt$, units watts (1 W = 1 J/s).
  • Also P = F v for a force pushing an object at speed v.
  • Same job done faster needs more power, not more work.
  • 1 horsepower ≈ 746 W.
  • A kilowatt-hour is energy: 1 kW $× 1 h = 3.6 × 10^6 J$.

Momentum and impulse

Momentum is mass in motion; impulse is the change a net force produces over time.
  • p = m v, a vector in kg·m/s.
  • Impulse $J = F_{avg} Δt = Δp$, units N·s.
  • On a force–time graph, impulse is the area under the curve.
  • Extending the contact time (airbags, bending knees) lowers the peak force for the same $Δp$.
  • Momentum is not the same as kinetic energy.

Collisions and conservation of momentum

With no external net force, total momentum is unchanged; collisions differ in what happens to kinetic energy.
  • $Σp_{before} = Σp_{after}$ for an isolated system, applied per axis.
  • Elastic: kinetic energy is also conserved.
  • Inelastic: some kinetic energy becomes heat, sound, or deformation.
  • Perfectly inelastic: objects stick together — $m_1 v_1 + m_2 v_2 = (m_1 + m_2) v_f$.
  • Explosion or recoil from rest: the fragments' momenta must sum to zero.
Using conservation of kinetic energy in an inelastic collision — only momentum is conserved there.
Using mgh with h as the distance along an incline instead of the vertical height.
Counting energy 'lost' to friction as destroyed — it becomes thermal energy.
Confusing impulse (N·s) with force (N), or momentum with kinetic energy.
Perpendicular force → zero work (normal force, centripetal force).
No time given but need a final speed? Use $W_{net} = ΔKE$.
Momentum is always conserved in a collision; kinetic energy only if it is elastic.
Longer collision time → smaller force (why crumple zones work).

Charge and Coulomb's law

Electric charge comes in two signs and exerts forces described by an inverse-square law.
  • Like charges repel; opposite charges attract. Charge is quantized in units of e = 1.6 × 10^-19 C.
  • Charge is conserved; objects charge by moving electrons, not by creating charge.
  • Coulomb's law: $F = k q_1 q_2 / r^2$, with $k = 8.99 × 10^9 N·m^2/C^2$.
  • Conductors let charge move freely; insulators hold it in place.
  • Same inverse-square form as gravity, but far stronger and able to attract or repel.

Electric field

The field E gives the force per unit charge a small test charge would feel.
  • E = F/q, units N/C (or V/m); it is a vector.
  • Point charge: $E = k Q / r^2$, pointing away from + and toward −.
  • Field lines leave + charges, enter − charges, and never cross.
  • Inside a conductor in electrostatic equilibrium, E = 0.
  • A uniform field exists between parallel charged plates.

Potential difference and current

Voltage drives charge through a circuit; current is the resulting flow rate.
  • Potential difference (voltage) is energy per unit charge, in volts (J/C).
  • Current $I = Δq / Δt$, in amperes (C/s); conventional current is the direction positive charge would move.
  • A battery's EMF maintains the potential difference that pushes current.
  • Power delivered: $P = I V = I^2 R = V^2 / R$, in watts.
  • Electrons drift slowly, but the electrical signal propagates near light speed.

Ohm's law and resistance

For many materials, current is proportional to voltage.
  • Ohm's law: V = I R, resistance R in ohms.
  • Resistance rises with length, falls with cross-sectional area, and depends on material and temperature.
  • An ohmic device has a straight-line I–V graph through the origin.
  • Resistors convert electrical energy to heat.
  • Ideal ammeter: zero resistance, in series. Ideal voltmeter: infinite resistance, in parallel.

Series and parallel circuits

How components are wired sets how voltage, current, and resistance distribute.
  • Series: one path. Same current everywhere; voltages add; $R_{total} = R_1 + R_2 +$ ... (increases).
  • Parallel: multiple paths. Same voltage across each branch; currents add; $1/R_{total} = 1/R_1 + 1/R_2 +$ ... (decreases).
  • Adding a parallel resistor lowers total resistance and raises total current from the battery.
  • A break in a series circuit stops all current; a break in one parallel branch leaves the others working.
  • At a junction, current in equals current out.
Saying charge is 'created' by rubbing — it is only transferred; total charge is conserved.
Adding parallel resistances directly instead of using the reciprocal rule.
Assuming adding resistors always increases total resistance — in parallel it decreases.
Swapping the ideal ammeter (0 ohm, series) and voltmeter (infinite ohm, parallel) placements.
Series adds resistance; parallel gives less than the smallest branch.
Same current in series; same voltage in parallel.
V = I R, and power has three faces: IV, $I^{2R}, V^2/R$.
Field lines: out of positive, into negative, never crossing.
Diagram
Series Parallel

Series circuit (same current everywhere, resistances add) vs parallel circuit (same voltage across each branch, resistances combine reciprocally).

Magnetic fields and forces

Moving charges create magnetic fields and feel forces in them.
  • Field lines run from N to S outside a magnet; isolated poles have never been found.
  • A current-carrying wire makes a circular magnetic field (right-hand rule: thumb = current, fingers curl = field).
  • Force on a moving charge: $F = q v B \sinθ$, perpendicular to both v and B.
  • A charge moving parallel to B feels no force; the force is maximum when v is perpendicular to B.
  • The magnetic force does no work — it changes direction, not speed, giving circular paths.

Electromagnetic induction

A changing magnetic flux through a loop induces a voltage.
  • Magnetic flux Φ = $B A \cosθ$ through a loop.
  • Faraday's law: induced EMF = $−ΔΦ/Δt$ — faster flux change means larger EMF.
  • Flux can change by varying B, the loop area, or the loop's orientation.
  • Lenz's law: the induced current opposes the change that produced it (energy conservation).
  • Basis of generators, transformers, microphones, and induction cooktops.

Motors, generators, transformers

The motor force and induction are two sides of the same electromagnetic coin.
  • Motor: current in a field feels a torque → rotation (electrical to mechanical).
  • Generator: rotating a coil in a field changes flux → induced AC EMF (mechanical to electrical).
  • Transformer: AC in a primary coil induces AC in a secondary; $V_s / V_p = N_s / N_p$.
  • A step-up transformer raises voltage and lowers current (ideal: power in = power out).
  • High-voltage transmission reduces $I^2 R$ losses in power lines.

Right-hand rules and charged-particle paths

A consistent hand convention predicts the direction of magnetic forces and fields.
  • For a moving positive charge: point fingers along v, curl toward B, thumb gives F (reverse for negative charge).
  • A charged particle entering a uniform field perpendicular to B moves in a circle.
  • Radius of that circle: r = m v / (q B) — faster or heavier particles curve less.
  • Parallel currents attract; antiparallel currents repel.
  • A velocity selector balances electric and magnetic forces: qE = qvB.

Solenoids and electromagnets

Coiling a current-carrying wire concentrates and shapes its magnetic field.
  • A solenoid produces a nearly uniform field inside, like a bar magnet's, and weak field outside.
  • Field strength grows with the current and with the number of turns per unit length.
  • Adding an iron core (an electromagnet) multiplies the field many times over.
  • Reversing the current reverses the north and south ends.
  • Electromagnets can be switched off, unlike permanent magnets — used in scrapyard cranes, relays, and maglev.
Expecting the magnetic force to speed a charge up — it does zero work.
Forgetting the $\sinθ$ in $F = q v B \sinθ$ (no force when v is parallel to B).
Ignoring the minus sign in Lenz's law — the induced current always opposes the flux change.
Thinking a steady magnetic field through a stationary loop induces a current — it does not.
Right-hand rule: fingers along velocity, curl to field, thumb is the force (flip for a negative charge).
Induction needs CHANGE — steady flux induces nothing.
Lenz = nature resists the change.
Step-up transformer trades current for voltage; power stays the same.

Defining simple harmonic motion

SHM is back-and-forth motion driven by a restoring force proportional to displacement.
  • Condition: F = −k x (the restoring force points back toward equilibrium and grows with displacement).
  • Amplitude A is the maximum displacement; in ideal SHM it does not affect the period.
  • Period T is the time for one full cycle; frequency f = 1/T, in hertz.
  • Speed is maximum at equilibrium; acceleration is maximum at the turning points.
  • The motion repeats exactly in the absence of damping.

Mass on a spring

A mass on an ideal spring is the archetypal harmonic oscillator.
  • $T = 2π \sqrt{m / k}$.
  • A stiffer spring (larger k) or a lighter mass gives a shorter period.
  • Quadrupling the mass doubles the period.
  • The equilibrium position is where the spring force balances gravity (for a vertical spring).
  • The period does not depend on the amplitude or on g.

The simple pendulum

A small-angle pendulum is also SHM, with a strikingly simple period.
  • $T = 2π \sqrt{L / g}$ for small swings.
  • Depends only on length and g — not on mass or (for small angles) amplitude.
  • Quadrupling the length doubles the period.
  • A pendulum runs slow where g is smaller (e.g. on the Moon).
  • Large swing angles break the $T = 2π \sqrt{L/g}$ approximation.

Energy in SHM

An oscillator continuously trades kinetic and potential energy while the total stays fixed.
  • Total mechanical energy = $0.5 k A^2$ for a mass–spring system.
  • All potential energy at the turning points; all kinetic energy at equilibrium.
  • Maximum speed $v_{max} = A \sqrt{k/m} = A ω$.
  • The energy is set by the amplitude and never exceeds its initial value.
  • At any point: $0.5 k x^2 + 0.5 m v^2 = 0.5 k A^2$.

Damping and resonance

Real oscillators lose energy, and driving them at the right frequency builds large amplitudes.
  • Damping (friction, air resistance) gradually reduces the amplitude while barely changing the period.
  • Light damping decays slowly; heavy damping can stop oscillation altogether (overdamped).
  • Every system has natural frequencies at which it 'likes' to oscillate.
  • Resonance: driving a system at a natural frequency transfers energy efficiently and grows the amplitude.
  • Resonance explains shattered wine glasses, pushed swings, and some bridge failures.
Thinking amplitude changes the period of ideal SHM — it does not.
Putting mass into the pendulum period — $T = 2π \sqrt{L/g}$ has no mass.
Saying acceleration is zero at the turning points — that is where it is largest.
Assuming damping quickly changes the period — it mainly shrinks the amplitude.
Restoring force proportional to displacement → SHM; period is amplitude-independent.
Spring period grows with mass; pendulum period grows with length.
Fast at the middle, momentarily stopped at the ends.
Drive at a natural frequency → resonance → big amplitude.

Wave basics

A wave transports energy through a medium without transporting the medium itself.
  • Transverse: particle motion is perpendicular to wave motion (string waves, light).
  • Longitudinal: particle motion is parallel to wave motion (sound).
  • Wave speed $v = f λ$.
  • Speed is set by the medium; changing frequency changes wavelength, not speed.
  • Amplitude relates to energy carried, not to speed.

Wave behavior

Waves reflect, refract, diffract, and interfere.
  • Reflection: a fixed end inverts a pulse; a free end does not.
  • Refraction: speed and wavelength change entering a new medium; frequency stays the same.
  • Diffraction: waves spread around edges and through gaps, most when the gap is near one wavelength.
  • Superposition: overlapping waves add displacement to displacement.
  • Constructive interference (in phase) builds amplitude; destructive (out of phase) cancels it.

Standing waves and resonance

Confined waves form fixed node/antinode patterns at special frequencies.
  • A standing wave is two identical waves traveling in opposite directions.
  • Nodes stay still; antinodes oscillate with maximum amplitude.
  • String fixed at both ends: $L = n λ / 2$, with $f_n = n v / (2L)$.
  • Resonance: driving a system at a natural frequency gives a large-amplitude response.
  • Instruments select harmonics through their length and boundary conditions.

Sound waves

Sound is a longitudinal pressure wave that needs a medium.
  • Speed in air ≈ 343 m/s at 20°C; faster in liquids and solids; zero in vacuum.
  • Pitch corresponds to frequency; loudness corresponds to amplitude/intensity.
  • Human hearing runs roughly 20 Hz to 20,000 Hz.
  • Intensity falls off with an inverse-square law from a point source.
  • Beats: two close frequencies give $f_{beat} = |f_1 − f_2|$.

The Doppler effect

Relative motion between a wave source and an observer shifts the observed frequency.
  • Source approaching → waves bunch up → higher observed frequency, shorter wavelength.
  • Source receding → waves stretch out → lower observed frequency.
  • Only motion along the line between source and observer matters.
  • Explains the pitch drop of a passing siren.
  • The same idea red-shifts light from receding galaxies.
Saying a wave speeds up if you shake the source faster — frequency up means wavelength down, speed fixed by the medium.
Confusing nodes (no motion) with antinodes (maximum motion).
Thinking the Doppler effect changes the wave's actual speed.
Assuming sound can travel through a vacuum.
$v = f λ$, and v belongs to the medium.
Fixed end flips the pulse; a both-ends-fixed string fits half-wavelengths.
Approaching source → higher pitch; receding → lower pitch.
Beat frequency is the difference of the two frequencies.

Reflection and mirrors

Light bouncing off surfaces obeys a simple angle rule and forms predictable images.
  • Law of reflection: angle of incidence = angle of reflection, measured from the normal.
  • Plane mirror: image is virtual, upright, same size, as far behind as the object is in front.
  • Concave (converging) mirror: real, inverted image when the object is beyond the focal point.
  • Convex (diverging) mirror: always a virtual, upright, reduced image.
  • Mirror equation: $1/f = 1/d_o + 1/d_i$; magnification $m = −d_i / d_o$.

Refraction and Snell's law

Light bends when it changes speed crossing between media.
  • Index of refraction $n = c / v_{medium}$; n ≥ 1, larger n means slower light.
  • Snell's law: $n_1 \sinθ_1 = n_2 \sinθ_2$.
  • Entering a denser (higher-n) medium bends light toward the normal.
  • Frequency and color do not change in refraction; speed and wavelength do.
  • A prism disperses white light because n varies slightly with wavelength.

Lenses

A lens forms images by refraction at its two curved surfaces.
  • Converging (convex) lens: real, inverted image for distant objects; magnifier for near objects.
  • Diverging (concave) lens: always virtual, upright, reduced.
  • Thin-lens equation matches the mirror equation: $1/f = 1/d_o + 1/d_i$.
  • A positive image distance means a real image on the far side; negative means a virtual image.
  • The focal length depends on the lens shape and the index of refraction.

Total internal reflection

Light going toward a less dense medium can be trapped entirely inside.
  • Occurs only going from higher n to lower n.
  • Happens when the angle of incidence exceeds the critical angle $θ_c$.
  • $\sinθ_c = n_2 / n_1$ (with $n_1$ > $n_2$).
  • This is how optical fibers guide light over long distances.
  • Also produces mirages and the sparkle of cut diamonds.

Wave optics and the electromagnetic spectrum

Interference and diffraction reveal light's wave nature; the spectrum orders it by wavelength.
  • A double slit produces bright/dark fringes from constructive/destructive interference.
  • A diffraction grating spreads light into sharp spectral orders.
  • Spectrum long to short wavelength: radio, microwave, infrared, visible, ultraviolet, X-ray, gamma.
  • All electromagnetic waves travel at c ≈ $3.0 × 10^8 m/s$ in vacuum.
  • Higher frequency means shorter wavelength and more energy per photon.
Measuring angles from the surface instead of from the normal.
Thinking a convex mirror or concave lens can make a real image — they cannot.
Saying refraction changes the light's frequency or color — only speed and wavelength change.
Expecting total internal reflection when going from low n to high n.
Toward the normal when entering a slower (higher-n) medium.
Convex mirror and concave lens: always virtual, upright, smaller.
$1/f = 1/d_o + 1/d_i$ for both mirrors and thin lenses.
TIR only high-n to low-n, past the critical angle.
Diagram
object image F

A converging (convex) lens: an object beyond the focal point (F) forms a real, inverted image on the far side where refracted rays converge.

Photons and the photoelectric effect

Light also behaves as discrete packets of energy called photons.
  • Photon energy $E = h f = h c / λ$, with h = 6.63 × 10^-34 J·s.
  • The photoelectric effect ejects electrons only if the frequency exceeds a threshold, no matter the intensity.
  • This showed energy is delivered in quanta, which wave theory alone could not explain.
  • Above threshold, brighter light ejects more electrons; higher frequency gives each electron more kinetic energy.
  • $KE_{max} = h f − W$, where W is the metal's work function.

Atomic energy levels and spectra

Electrons in an atom occupy discrete energy levels, and light carries the difference when they jump.
  • An electron absorbs a photon to jump up and emits a photon when it falls down.
  • The photon energy equals the difference between the two levels: $E_{photon} = E_{high} − E_{low}$.
  • Each element has a unique line spectrum — a fingerprint of its energy levels.
  • Emission spectra are bright lines; absorption spectra are dark lines at the same wavelengths.
  • Ground state is the lowest level; higher levels are excited states.

The nucleus and radioactivity

Unstable nuclei decay by emitting particles or photons.
  • Alpha decay emits a helium-4 nucleus; mass number drops by 4, atomic number by 2.
  • Beta-minus decay emits an electron; a neutron becomes a proton, so atomic number rises by 1.
  • Gamma decay emits a high-energy photon; the nucleus loses energy but not particles.
  • Half-life is the time for half of a sample to decay; it is constant for a given isotope.
  • Mass number and charge are conserved when balancing a nuclear equation.

Mass–energy equivalence

Mass and energy are two forms of the same thing.
  • $E = m c^2$; because $c^2$ is enormous, a tiny mass change releases huge energy.
  • The mass of a nucleus is slightly less than its separate nucleons — the missing 'mass defect' is binding energy.
  • Nuclear reactions release far more energy per kilogram than chemical reactions.
  • Energy is still conserved overall; some rest mass simply converts to kinetic energy and radiation.
  • Units: 1 u of mass corresponds to about 931 MeV of energy.

Fission and fusion

Both processes move nuclei toward the peak of the binding-energy curve, releasing energy.
  • Fission splits a heavy nucleus (e.g. uranium-235) into lighter fragments plus neutrons.
  • A chain reaction sustains fission when released neutrons trigger further splits.
  • Fusion joins light nuclei (e.g. hydrogen isotopes) into a heavier one; it powers the Sun.
  • Fusion releases more energy per nucleon but needs extreme temperature and pressure.
  • Iron-56 sits near the peak of binding energy per nucleon, so it releases energy in neither process.
Thinking brighter light always ejects photoelectrons — the frequency must first exceed the threshold.
Saying an electron can occupy any energy in an atom — the levels are discrete.
Forgetting to conserve both mass number and atomic number in nuclear equations.
Confusing fission (splitting heavy nuclei) with fusion (joining light nuclei).
Photon energy: E = h f — higher frequency, more energetic.
A spectral line is a jump between two fixed energy levels.
Alpha: −4 mass, −2 charge. Beta-minus: +1 charge, same mass number.
Mass defect = binding energy through $E = m c^2$.
Practice Question Bank — 468 questions
Unit 1: Measurement, Vectors & Motion Basics (184)

  1. The SI unit of mass is the:

    • gram
    • kilogram
    • newton
    • slug

    The kilogram is the SI base unit of mass; the gram is a smaller metric unit.

  2. Which of these is a derived unit rather than a base unit?

    • newton
    • meter
    • second
    • kilogram

    The newton is defined as kg·m/s², a combination of base units.

  3. Which of these is an SI base unit?

    • newton
    • joule
    • watt
    • ampere

    The ampere (electric current) is one of the seven base units; the others listed are derived.

  4. A student solves F = ma with m = 5 kg and a = 10 m/s² and writes only "F = 50". Why does the answer lose credit?

    • The formula was wrong
    • The calculation is wrong
    • The unit (newtons) is missing
    • F should be 15

    5 × 10 = 50 is correct, but a physics answer needs its unit: 50 N.

  5. Which prefix stands for 10^12?

    • milli
    • tera
    • kilo
    • pico

    tera = 10^12; pico is 10^-12.

  6. Which prefix stands for 10^3?

    • kilo
    • deci
    • tera
    • milli

    kilo means one thousand, 10^3.

  7. A megawatt is:

    • 10^9 W
    • 10^3 W
    • 10^-9 W
    • 10^6 W

    mega = 10^6.

  8. 5.0 ms (milliseconds) expressed in seconds is:

    • 5.0 × 10^3 s
    • 5.0 × 10^-6 s
    • 5.0 × 10^-3 s
    • 5.0 × 10^-2 s

    milli = 10^-3.

  9. 3.5 km expressed in meters is:

    • 3.5 × 10^-3 m
    • 3.5 × 10^3 m
    • 3.5 × 10^6 m
    • 3.5 × 10^2 m

    kilo = 10^3, so 3.5 km = 3500 m.

  10. 250 nm expressed in meters is:

    • 2.5 × 10^-7 m
    • 2.5 × 10^-9 m
    • 2.5 × 10^-6 m
    • 2.5 × 10^7 m

    250 × 10^-9 m = 2.5 × 10^-7 m.

  11. The average distance from the Sun to Mars is 227 800 000 000 m. In scientific notation this is:

    • 2.278 × 10^12 m
    • 22.78 × 10^10 m
    • 2.278 × 10^10 m
    • 2.278 × 10^11 m

    Move the decimal 11 places left: 2.278 × 10^11 m.

  12. 0.000 000 103 m written in scientific notation is:

    • 1.03 × 10^-6 m
    • 1.03 × 10^7 m
    • 1.03 × 10^-7 m
    • 10.3 × 10^-8 m

    Move the decimal 7 places right: 1.03 × 10^-7 m.

  13. 2 003 000 000 m written in scientific notation is:

    • 2.003 × 10^10 m
    • 2.003 × 10^9 m
    • 2.003 × 10^8 m
    • 2.03 × 10^9 m

    Nine places to the left gives 2.003 × 10^9 m; keep the zeros between digits.

  14. What is the order of magnitude of 72 meters per second?

    • 10^2
    • 10^1
    • 10^0
    • 10^4

    72 = 7.2 × 10^1, and 7.2 is closer to 10 than to 1, so the order of magnitude is 10^2.

  15. What is the order of magnitude of 8756 kg?

    • 10^3 kg
    • 10^2 kg
    • 10^5 kg
    • 10^4 kg

    8756 = 8.756 × 10^3, and 8.756 is closer to 10 than to 1, so it is on the order of 10^4 kg.

  16. What is the order of magnitude of 8.2 × 10^-9 m?

    • 10^-9 m
    • 10^-10 m
    • 10^-8 m
    • 10^-7 m

    8.2 is closer to 10 than 1, so round up to 10 × 10^-9 = 10^-8 m.

  17. Compute 4 × 10^8 m + 3 × 10^8 m.

    • 7 × 10^16 m
    • 7 × 10^8 m
    • 12 × 10^8 m
    • 7 × 10^0 m

    Same power of 10, so add the M values and keep the exponent.

  18. Compute 5.80 × 10^9 s + 3.20 × 10^8 s.

    • 6.12 × 10^9 s
    • 9.00 × 10^9 s
    • 6.12 × 10^8 s
    • 9.00 × 10^17 s

    Rewrite 3.20 × 10^8 as 0.320 × 10^9, then 5.80 + 0.320 = 6.12.

  19. Compute 4.87 × 10^-6 m − 1.93 × 10^-6 m.

    • 2.94 × 10^-12 m
    • 6.80 × 10^-6 m
    • 2.94 × 10^6 m
    • 2.94 × 10^-6 m

    Same exponent, so subtract: 4.87 − 1.93 = 2.94.

  20. Compute 3.14 × 10^-5 kg + 9.36 × 10^-5 kg.

    • 12.50 × 10^-5 kg
    • 1.25 × 10^-10 kg
    • 1.25 × 10^-4 kg
    • 1.25 × 10^-5 kg

    3.14 + 9.36 = 12.50 × 10^-5, which in proper scientific notation is 1.250 × 10^-4.

  21. Compute 8.12 × 10^7 g − 6.20 × 10^6 g.

    • 1.92 × 10^7 g
    • 7.50 × 10^7 g
    • 1.92 × 10^1 g
    • 7.50 × 10^6 g

    6.20 × 10^6 = 0.620 × 10^7, and 8.12 − 0.620 = 7.50.

  22. Compute (6.2 × 10^18 m)(4.7 × 10^-10 m).

    • 2.9 × 10^9 m²
    • 2.9 × 10^28 m
    • 2.9 × 10^8 m²
    • 10.9 × 10^8 m²

    6.2 × 4.7 = 29.14 and 18 + (−10) = 8, so 29.14 × 10^8 = 2.9 × 10^9 m².

  23. Compute (5.6 × 10^-7 m) / (2.8 × 10^-12 s).

    • 2.0 × 10^-5 m/s
    • 2.0 × 10^-19 m/s
    • 2.0 × 10^6 m/s
    • 2.0 × 10^5 m/s

    5.6 / 2.8 = 2.0 and −7 − (−12) = +5.

  24. Compute (8 × 10^6 m³) / (2 × 10^-3 m²).

    • 4 × 10^3 m
    • 4 × 10^-9 m
    • 4 × 10^9 m
    • 16 × 10^3 m

    8 / 2 = 4 and 6 − (−3) = 9; the units m³/m² leave m.

  25. Compute (6.5 × 10^5 kg) / (3.4 × 10^3 m³).

    • 1.9 × 10^8 kg/m³
    • 1.9 × 10^2 kg/m³
    • 1.9 × 10^-2 kg/m³
    • 2.2 × 10^9 kg/m³

    6.5 / 3.4 = 1.9 and 5 − 3 = 2; this is a density.

  26. Which is the most likely mass of a high school student?

    • 60 kg
    • 6 kg
    • 250 kg
    • 600 kg

    A typical teenager has a mass around 60 kg.

  27. The height of an average adult is approximately:

    • 1.75 cm
    • 1.75 mm
    • 1.75 km
    • 1.75 m

    Adults are a little under 2 m tall.

  28. The length of a high school physics classroom is closest to:

    • 10^-1 m
    • 10^3 m
    • 10^1 m
    • 10^5 m

    A classroom is on the order of 10 m long.

  29. Which measurement of an average classroom door is closest to 1 meter?

    • thickness
    • width
    • height
    • surface area

    A door is roughly 0.9 m wide; its height is about 2 m and its thickness a few cm.

  30. The weight of an apple is closest to:

    • 1 N
    • 100 N
    • 10 N
    • 0.01 N

    An apple has a mass of about 0.1 kg, so its weight is about 1 N.

  31. The measurement 24.962 s is precise to the nearest:

    • 0.01 s
    • 0.1 s
    • 1 s
    • 0.001 s

    The last digit sits in the thousandths place.

  32. How many significant figures are in 0.00420 kg?

    • 2
    • 5
    • 3
    • 6

    Leading zeros do not count; 4, 2, and the trailing 0 after the decimal do.

  33. How many significant figures are in 3000 K (no decimal point)?

    • 4
    • 1
    • 2
    • 3

    Trailing zeros with no decimal point are not significant.

  34. The average of several measurements of a mass is 48.60 g and the accepted value is 48.75 g. The percent error is about:

    • 0.31%
    • 0.15%
    • 3.1%
    • 0.0031%

    |48.60 − 48.75| / 48.75 × 100% = 0.15 / 48.75 × 100% ≈ 0.31%.

  35. A thermometer reads 5°C in an ice-and-water mixture that is really at 0°C. This is what kind of error?

    • random
    • percent
    • significant
    • systematic

    Every reading is off the same way, which is systematic error; calibrating the thermometer fixes it.

  36. Random errors in a measurement are best reduced by:

    • using a bigger unit
    • calibrating once
    • averaging many measurements
    • rounding to one digit

    Random fluctuations tend to cancel in an average of many trials.

  37. Which quantity is a scalar?

    • an airplane traveling 300 m/s east
    • the temperature outside is 20°C
    • a 10 N force downward
    • a displacement of 3 m north

    Temperature has magnitude only; the others include a direction.

  38. Which quantity is a vector?

    • momentum
    • mass
    • energy
    • density

    Momentum has both magnitude and direction.

  39. In a vector diagram, the length of the arrow represents the vector's:

    • direction
    • sign
    • unit only
    • magnitude

    Arrow length is proportional to magnitude; the arrowhead shows direction.

  40. Two forces are called concurrent when they:

    • act on the same point at the same time
    • act along the same line in the same direction
    • have equal magnitudes and opposite directions
    • act at right angles to each other at one instant

    Concurrent means acting together on one point.

  41. Two concurrent forces of 3 N and 4 N act at 90° to each other. The magnitude of the resultant is:

    • 7 N
    • 5 N
    • 1 N
    • 12 N

    √(3² + 4²) = √25 = 5 N.

  42. An object is displaced 12 meters to the right and then 16 meters upward. The magnitude of its total displacement is:

    • 20 m
    • 28 m
    • 4.0 m
    • 1.3 m

    √(12² + 16²) = √400 = 20 m.

  43. An object is displaced 3 meters west and then 4 meters south. The magnitude of its displacement is:

    • 7 m
    • 1 m
    • 12 m
    • 5 m

    √(3² + 4²) = 5 m.

  44. A girl swims directly across a 15 m wide stream and ends up 15 m downstream. The magnitude of her displacement is closest to:

    • 30 m
    • 15 m
    • 21 m
    • 17 m

    √(15² + 15²) = 15√2 ≈ 21 m.

  45. A boat heads directly east across a river at 12 m/s while the current flows due south at 5.0 m/s. The magnitude of the boat's resultant velocity is:

    • 17 m/s
    • 13 m/s
    • 7.0 m/s
    • 8.5 m/s

    √(12² + 5.0²) = √169 = 13 m/s.

  46. Two forces of 100 N and 25 N act concurrently on an object. Their resultant could be:

    • 100 N
    • 0 N
    • 50 N
    • 150 N

    The resultant must lie between 100 − 25 = 75 N and 100 + 25 = 125 N; only 100 N fits.

  47. The resultant of two concurrent forces is a minimum when the angle between them is:

    • 0°
    • 45°
    • 90°
    • 180°

    Opposite forces cancel as much as possible.

  48. As the angle between two concurrent forces of 5.0 N and 7.0 N increases from 0° to 180°, the magnitude of their resultant changes from:

    • 2.0 N to 12 N
    • 0 N to 12 N
    • 12 N to 2.0 N
    • 12 N to 0 N

    At 0° the forces add (12 N); at 180° they subtract (2.0 N).

  49. Two 10-N concurrent forces have a resultant of zero. The angle between them must be:

    • 0°
    • 180°
    • 45°
    • 90°

    Only equal and opposite forces sum to zero.

  50. A force of 3 N and a force of 5 N act concurrently to produce a resultant of 8 N. The angle between them must be:

    • 0°
    • 60°
    • 90°
    • 180°

    8 = 3 + 5 only when the forces point the same way.

  51. Two concurrent forces have a maximum resultant of 45 N and a minimum resultant of 5 N. What are their magnitudes?

    • 5 N and 9 N
    • 0 N and 45 N
    • 22.5 N and 22.5 N
    • 20. N and 25 N

    A + B = 45 and A − B = 5, so A = 25 N and B = 20 N.

  52. A 6.0 N force and an 8.0 N force act concurrently at 90° to each other. The magnitude of the resultant is:

    • 14 N
    • 2.0 N
    • 10 N
    • 7.0 N

    √(6² + 8²) = 10 N.

  53. Which set of three concurrent forces could NOT produce equilibrium?

    • 2 N, 2 N, 2 N
    • 1 N, 3 N, 5 N
    • 3 N, 4 N, 5 N
    • 4 N, 4 N, 5 N

    For equilibrium the largest force must not exceed the sum of the other two; 1 + 3 < 5.

  54. A 60. N force acts 30° above the horizontal. Its horizontal component is closest to:

    • 52 N
    • 30 N
    • 60 N
    • 35 N

    60 cos 30° ≈ 52 N.

  55. A 100 N force acts 30° above the horizontal. Its vertical component is:

    • 87 N
    • 100 N
    • 30 N
    • 50 N

    100 sin 30° = 50 N.

  56. A force of 28.0 N is applied to a lawn-roller handle held 45° above the horizontal. The force moving the roller forward is:

    • 28.0 N
    • 14.0 N
    • 19.8 N
    • 39.6 N

    28.0 cos 45° ≈ 19.8 N.

  57. A resultant force of 10. N is made of two perpendicular components. If one component is 6.0 N, the other is:

    • 16 N
    • 8.0 N
    • 4.0 N
    • 6.0 N

    √(10² − 6²) = 8.0 N.

  58. A block of weight F_g rests on a 30° incline. The component of its weight parallel to the incline is:

    • 0.50 F_g
    • 0.87 F_g
    • F_g
    • 1.7 F_g

    F_g sin 30° = 0.50 F_g.

  59. A lawn mower is pushed with a constant force along its handle. As the angle between the handle and the horizontal increases, the horizontal component of the force:

    • increases
    • remains the same
    • becomes zero at 45°
    • decreases

    Horizontal component = F cosθ, and cosθ decreases as θ increases.

  60. A 300. N force acts on a shovel handle at 60° above the ground. Its vertical component is closest to:

    • 150 N
    • 300 N
    • 260 N
    • 520 N

    300 sin 60° ≈ 260 N.

  61. A runner completes one full lap of a 400 m track. Her distance and displacement are:

    • 0 m and 400 m
    • 400 m and 0 m
    • 400 m and 400 m
    • 0 m and 0 m

    She travels 400 m along the path but ends where she started.

  62. A car drives 30 km east and then 40 km north. Its displacement has a magnitude of:

    • 50 km
    • 70 km
    • 10 km
    • 35 km

    √(30² + 40²) = 50 km, while the distance is 70 km.

  63. A car travels 120 m in 8.0 s. Its average speed is:

    • 0.067 m/s
    • 960 m/s
    • 112 m/s
    • 15 m/s

    d/t = 120 / 8.0 = 15 m/s.

  64. A motorist drives 60 km in the first hour and 40 km in the second hour. Her average speed is:

    • 100 km/h
    • 20 km/h
    • 50 km/h
    • 60 km/h

    Total distance 100 km divided by total time 2 h.

  65. A student walks 40 m east in 20 s and then 40 m west in 20 s. Her average velocity for the whole trip is:

    • 2.0 m/s east
    • 0 m/s
    • 2.0 m/s west
    • 80 m/s

    The net displacement is zero, so the average velocity is zero.

  66. For the same trip (40 m east then 40 m west in 40 s total), her average speed is:

    • 2.0 m/s
    • 0 m/s
    • 1.0 m/s
    • 80 m/s

    Average speed = total distance / total time = 80 m / 40 s.

  67. A hiker's displacement is 150 m north in 30 s. His average velocity is:

    • 5.0 m/s
    • 4500 m/s north
    • 0.20 m/s north
    • 5.0 m/s north

    A velocity needs a direction: 150 / 30 = 5.0 m/s north.

  68. Which quantity includes a direction?

    • speed
    • distance
    • velocity
    • time

    Velocity is a vector; speed, distance, and time are scalars.

  69. The reading on a car's speedometer at one moment is the car's:

    • average speed
    • instantaneous speed
    • displacement
    • average velocity

    The speedometer gives speed at that instant, with no direction.

  70. An object's velocity is −4 m/s. This means it is:

    • moving in the positive direction at 4 m/s
    • moving in the negative direction at −4 m/s speed
    • moving in the negative direction at 4 m/s
    • at rest at a position of −4 m from the origin

    The sign gives direction; the speed is 4 m/s.

  71. A cyclist rides at a steady 8.0 m/s for 15 s. The distance covered is:

    • 1.9 m
    • 23 m
    • 8.0 m
    • 120 m

    d = vt = 8.0 × 15 = 120 m.

  72. How long does it take to travel 300 km at a constant 60 km/h?

    • 0.20 h
    • 18 000 h
    • 5.0 h
    • 360 h

    t = d / v = 300 / 60 = 5.0 h.

  73. Which statement about distance and displacement is always true?

    • The magnitude of the displacement is never greater than the distance
    • The distance traveled is never greater than the magnitude of the displacement
    • The magnitude of the displacement equals the distance on any round trip
    • Distance is a vector quantity that includes the direction of motion

    The straight line from start to finish is the shortest path.

  74. The dot product of two vectors is:

    • a scalar
    • a vector
    • always zero
    • always negative

    A·B = AB cosθ is a single number; the cross product is the vector one.

  75. The cross product of two parallel vectors is:

    • a maximum
    • a scalar
    • twice the dot product
    • zero

    |A×B| = AB sinθ, and sin 0° = 0.

  76. To subtract vector B from vector A, you can:

    • add a vector with B's length pointing the same way as A
    • add B to A after rotating B by 90° counterclockwise
    • multiply A by the length of B and reverse its direction
    • add a vector with B's length pointing the opposite way

    A − B = A + (−B).

  77. A girl walks 10. meters north to a drinking fountain and then 30. meters south to an art classroom. Her total displacement from the start is:

    • 20. m north
    • 40. m south
    • 20. m south
    • 40. m north

    Taking north as positive: +10 − 30 = −20, so the displacement is 20. m south (the distance walked is 40. m).

  78. A projectile is fired at 120. m/s at an angle θ above the horizontal. If its initial horizontal speed is 55 m/s, the angle θ is approximately:

    • 27°
    • 63°
    • 46°
    • 55°

    v_x = v cosθ, so cosθ = 55/120 = 0.458 and θ ≈ 63°.

  79. A student walks four blocks south, five blocks west, and four more blocks south. Compared with the distance she walks, the magnitude of her displacement from home to school is:

    • less
    • greater
    • the same
    • impossible to determine

    She walks 13 blocks, but her displacement is √(8² + 5²) ≈ 9.4 blocks; the straight line is always the shortest path.

  80. A vector makes an angle θ with the horizontal. Its horizontal and vertical components have equal magnitudes when θ is:

    • 30°
    • 60°
    • 90°
    • 45°

    A cosθ = A sinθ only when tanθ = 1, which is at 45°.

  81. A car travels 90. m due north in 15 s, then turns around and travels 40. m due south in 5.0 s. The magnitude of its average velocity for the 20-s interval is:

    • 6.5 m/s
    • 5.0 m/s
    • 2.5 m/s
    • 0.40 m/s

    Displacement = 90. − 40. = 50. m over 20 s = 2.5 m/s. (6.5 m/s would be the average speed, 130 m / 20 s.)

  82. In a 4.0-km race, a runner covers the four 1.0-km sections in 5.9, 6.2, 6.3, and 6.0 minutes. Her average speed for the race is about:

    • 0.25 km/min
    • 0.16 km/min
    • 6.1 km/min
    • 24 km/min

    Total time = 24.4 min, so 4.0 km / 24.4 min ≈ 0.16 km/min.

  83. A golf ball is hit at 15 m/s at 35° above the horizontal. The vertical component of its initial velocity is:

    • 8.6 m/s
    • 12 m/s
    • 15 m/s
    • 21 m/s

    15 sin 35° ≈ 8.6 m/s (the horizontal component is 15 cos 35° ≈ 12 m/s).

  84. The average speed of a runner in a 400.-m race is 8.0 m/s. How long did the race take?

    • 5.0 s
    • 0.020 s
    • 3200 s
    • 50. s

    t = d / v = 400. / 8.0 = 50. s.

  85. An object travels for 8.00 s at an average speed of 160. m/s. The distance traveled is:

    • 20.0 m
    • 152 m
    • 1280 m
    • 168 m

    d = vt = 160. × 8.00 = 1280 m.

  86. An object moves a distance of 10 m in 5 s. Its average speed is:

    • 0.50 m/s
    • 2.0 m/s
    • 5.0 m/s
    • 50 m/s

    v = d/t = 10 / 5 = 2.0 m/s.

  87. A moving body must undergo a change of:

    • position
    • velocity
    • acceleration
    • direction

    Motion means position changes with time; a body moving at constant velocity has no change in velocity or acceleration, and a straight-line mover has no change in direction.

  88. What is the largest number of components a single force can be resolved into?

    • two
    • three
    • four
    • an unlimited number

    A vector can be replaced by any number of vectors that add up to it; two perpendicular components are just the most useful choice.

  89. A man walks 17 m east and then 17 m south. The magnitude of his displacement is:

    • 17 m
    • 34 m
    • 24 m
    • 0 m

    √(17² + 17²) = 17√2 ≈ 24 m.

  90. A worker pulls a cart with a rope. The worker's pull can best be described as a force having:

    • magnitude only
    • both magnitude and direction
    • direction only
    • neither magnitude nor direction

    Force is a vector, so a complete description needs a size and a direction.

  91. Which of these is a vector quantity?

    • force
    • mass
    • time
    • distance

    Force has direction; mass, time, and distance are scalars.

  92. A 5.0 N force could have perpendicular components of:

    • 2.0 N and 3.0 N
    • 1.0 N and 4.0 N
    • 5.0 N and 5.0 N
    • 3.0 N and 4.0 N

    The components must satisfy √(A² + B²) = 5.0, and 3.0² + 4.0² = 5.0².

  93. Two 10.0 N forces act concurrently on a point at an angle of 180° to each other. The magnitude of their resultant is:

    • 10.0 N
    • 14.1 N
    • 0.00 N
    • 20.0 N

    Equal forces pointing in opposite directions cancel.

  94. As the angle between two concurrent forces of 10 N and 12 N changes from 180° to 0°, the magnitude of their resultant changes from:

    • 22 N to 2.0 N
    • 2.0 N to 22 N
    • 0 N to 22 N
    • 2.0 N to 15.6 N

    At 180° the forces subtract (12 − 10 = 2.0 N); at 0° they add (22 N).

  95. Two concurrent forces act at right angles. If one force is 40 N and their resultant is 50 N, the other force is:

    • 30 N
    • 10 N
    • 20 N
    • 64 N

    √(50² − 40²) = √900 = 30 N.

  96. A 5.0 N force and a 12 N force act at right angles on point P. The magnitude of their resultant is:

    • 7.0 N
    • 17 N
    • 60 N
    • 13 N

    √(5.0² + 12²) = √169 = 13 N.

  97. Two 30.-N forces act concurrently on an object. The resultant is also 30. N when the angle between the forces is:

    • 60°
    • 90°
    • 120°
    • 180°

    With equal forces, R = 2F cos(θ/2); R = F requires cos(θ/2) = 0.5, so θ = 120°.

  98. A constant force F is applied to a mass on a frictionless horizontal surface at an angle θ above the horizontal. As θ increases, the mass's horizontal acceleration:

    • increases
    • decreases
    • stays the same
    • becomes negative

    The horizontal component F cosθ shrinks as θ grows, so a = F cosθ / m decreases.

  99. A skater speeds up uniformly from 2.0 m/s to 7.0 m/s over a distance of 12 m. What is the magnitude of her acceleration?

    • 1.9 m/s²
    • 0.42 m/s²
    • 3.8 m/s²
    • 2.2 m/s²

    No time is given, so use v² = v₀² + 2ad: 7.0² = 2.0² + 2a(12), so 49 = 4 + 24a and a = 45 ÷ 24 ≈ 1.9 m/s².

  100. A wagon's speed increases uniformly from 2.5 m/s to 9.0 m/s in 3.0 s as it rolls down a hill. What is the magnitude of its acceleration?

    • 2.2 m/s²
    • 3.8 m/s²
    • 6.5 m/s²
    • 0.46 m/s²

    a = Δv ÷ t = (9.0 − 2.5) m/s ÷ 3.0 s = 6.5 ÷ 3.0 ≈ 2.2 m/s². 6.5 m/s² forgets to divide by the time.

  101. An object moving at 4.0 m/s accelerates uniformly at 2.0 m/s² in its direction of motion for 5.0 m. What is its final speed?

    • 6.0 m/s
    • 14 m/s
    • 4.5 m/s
    • 36 m/s

    v² = v₀² + 2ad = 4.0² + 2(2.0)(5.0) = 16 + 20 = 36, so v = √36 = 6.0 m/s. 36 m/s forgets the square root.

  102. Which statement about an object with zero acceleration is true?

    • The object may be in motion.
    • The object must be at rest.
    • The object must be slowing down.
    • The object's velocity must be changing.

    Zero acceleration means the velocity isn't changing, not that it is zero. The object can be at rest or moving at a constant velocity.

  103. Acceleration is a vector quantity that represents the time-rate of change in:

    • velocity
    • position
    • distance
    • mass

    Acceleration = Δv ÷ Δt. The rate of change of position is velocity, not acceleration.

  104. On a velocity–time graph, an object moving with constant velocity appears as:

    • a horizontal straight line
    • a straight line sloping upward
    • a curve that gets steeper
    • a vertical line

    Constant velocity means v doesn't change as time passes, so every point sits at the same height: a horizontal line.

  105. On a distance–time graph, an object in uniform motion appears as:

    • a horizontal line with a slope of zero
    • a straight line with constant slope
    • a curve with a steadily increasing slope
    • a curved line that levels off over time

    Equal distances are covered in equal times, so distance grows linearly with time. A horizontal d–t line would mean the object isn't moving.

  106. The slope of a distance–time graph represents the object's:

    • velocity
    • acceleration
    • distance traveled
    • mass

    Slope = rise/run = Δd/Δt, which is velocity. The slope of a v–t graph is acceleration.

  107. A position–time graph is a straight line with a negative slope. The object is:

    • speeding up steadily in the negative direction
    • moving at constant speed in the negative direction
    • moving at constant speed in the positive direction
    • at rest at a negative position

    A straight line means constant velocity; the negative slope means that velocity points in the negative direction.

  108. An object moves at a constant 6.0 m/s for 5.0 s. On its v–t graph, the area under the line equals:

    • 30 m, the distance traveled
    • 1.2 m/s², the acceleration
    • 30 m/s, the final speed
    • 0 m, because the velocity is constant

    The area under a v–t graph is distance: a rectangle 6.0 m/s tall and 5.0 s wide has area 30 m.

  109. A distance–time graph is a straight line from 0 m at t = 0 to 40. m at t = 8.0 s. The object's speed is:

    • 5.0 m/s
    • 320 m/s
    • 0.20 m/s
    • 40. m/s

    Speed = slope = 40. m ÷ 8.0 s = 5.0 m/s. 0.20 m/s divides the wrong way.

  110. A car slows uniformly from 20. m/s to 5.0 m/s in 3.0 s. Its acceleration is:

    • −5.0 m/s²
    • 5.0 m/s²
    • −8.3 m/s²
    • −15 m/s²

    a = (v_f − v_i)/t = (5.0 − 20.)/3.0 = −5.0 m/s². The minus sign shows the acceleration opposes the motion. −15 forgets to divide by time.

  111. Deceleration means that an object's acceleration:

    • points opposite to its velocity
    • is negative
    • is zero
    • points the same way as its velocity

    Deceleration is slowing down, which happens whenever a and v point in opposite directions. A negative acceleration only slows the object if the velocity is positive.

  112. An object has a negative velocity and a negative acceleration. It is:

    • slowing down while moving in the negative direction
    • speeding up while moving in the positive direction
    • moving at constant speed in the negative direction
    • speeding up in the negative direction

    Velocity and acceleration point the same way (both negative), so the speed increases, in the negative direction.

  113. A car rounds a curve at a constant 15 m/s. Is it accelerating?

    • Yes, because the direction of its velocity is changing
    • No, because its speed stays constant at 15 m/s throughout the turn
    • No, because acceleration only occurs when the car speeds up
    • Yes, but only while the driver is pressing the gas pedal

    Velocity is a vector. Changing its direction changes the velocity, and any change in velocity is acceleration.

  114. A cart starts from rest and accelerates uniformly at 3.0 m/s² for 6.0 s. Its final speed is:

    • 18 m/s
    • 9.0 m/s
    • 2.0 m/s
    • 54 m/s

    v_f = v_0 + at = 0 + (3.0)(6.0) = 18 m/s.

  115. A car starts from rest and accelerates uniformly at 2.0 m/s² for 5.0 s. How far does it travel?

    • 25 m
    • 10. m
    • 50. m
    • 5.0 m

    From rest, d = ½at² = ½(2.0)(5.0)² = 25 m. 50 m forgets the ½; 10 m is the final speed's number.

  116. Which constant-acceleration equation should you use when the time is not given?

    • v_f² = v_0² + 2ad
    • v_f = v_0 + at
    • d = v_0t + ½at²
    • a = Δv/Δt

    v_f² = v_0² + 2ad is the only one of these with no t in it.

  117. The SI unit of acceleration is:

    • m/s²
    • m/s
    • m²/s
    • N/s

    Acceleration is change in velocity (m/s) per second, so its unit is (m/s)/s = m/s².

  118. A position–time graph that curves (is not a straight line) shows that the object's:

    • velocity is changing
    • velocity is constant
    • distance is not changing
    • acceleration is zero

    A curve means the slope, which is the velocity, changes from point to point: nonuniform motion.

  119. On a curved position–time graph, the instantaneous velocity at a point is:

    • the slope of the tangent line at that point
    • the slope of the line from the start to that point
    • the area under the curve up to that point
    • the height of the curve at that point

    The tangent slope gives the velocity at that instant; the slope of a line joining two points gives the average velocity between them.

  120. On a position–time graph, the average velocity for a whole trip is the slope of:

    • the tangent drawn to the curve at its starting point
    • the steepest tangent line anywhere along the curve
    • the tangent drawn to the curve at its ending point
    • the straight line joining the starting and ending points

    Average velocity = total displacement ÷ total time, which is the slope of the straight line (secant) between the endpoints.

  121. At one point on a position–time curve the tangent line is horizontal. At that instant the object is:

    • momentarily at rest
    • moving at its top speed
    • accelerating at zero
    • moving in the negative direction

    A horizontal tangent has zero slope, so the instantaneous velocity is zero: the object has stopped, often just as it turns around.

  122. The slope of a velocity–time graph represents:

    • acceleration
    • velocity
    • distance traveled
    • displacement

    Slope = Δv/Δt, which is acceleration. The area under a v–t graph is distance.

  123. A v–t graph is a straight line from 0 m/s at t = 0 to 12 m/s at t = 4.0 s. The object's acceleration and distance traveled are:

    • 3.0 m/s² and 24 m
    • 3.0 m/s² and 48 m
    • 48 m/s² and 24 m
    • 0.33 m/s² and 12 m

    Acceleration = slope = 12 ÷ 4.0 = 3.0 m/s². Distance = area of the triangle = ½ × 4.0 s × 12 m/s = 24 m (48 m forgets the ½).

  124. Compared with y = mx + b, the equation v = v_0 + at shows that on a v–t graph:

    • the slope is a and the vertical intercept is v_0
    • the slope is v_0 and the intercept is a
    • the slope is t and the intercept is v
    • the slope is v and the intercept is t

    Write it as v = (a)t + v_0: v plays y, t plays x, a is the slope m, and v_0 is the intercept b.

  125. The height of a 30-story building is approximately:

    • 10² m
    • 10¹ m
    • 10³ m
    • 10⁰ m

    Each story is about 3 m, so 30 × 3 m ≈ 90 m, which is closest to 10² m.

  126. The length of a football field is closest to:

    • 1000 dm
    • 1000 cm
    • 1000 km
    • 1000 mm

    A football field is about 100 m. 1000 dm = 100 m; 1000 cm is only 10 m and 1000 mm is 1 m.

  127. What is the approximate width of a person's little finger?

    • 0.01 m
    • 1 m
    • 0.1 m
    • 0.001 m

    A little finger is about 1 cm wide, and 1 cm = 0.01 m.

  128. What is the approximate mass of a pencil?

    • 5.0 × 10⁻³ kg
    • 5.0 × 10⁻¹ kg
    • 5.0 × 10⁰ kg
    • 5.0 × 10¹ kg

    A pencil is about 5 g. Since 1000 g = 1 kg, 5 g = 0.005 kg = 5.0 × 10⁻³ kg.

  129. Two displacements of 3 m and 4 m are added. What are the largest and smallest possible resultants?

    • 7 m and 1 m
    • 7 m and 5 m
    • 5 m and 1 m
    • 12 m and 0 m

    Largest when they point the same way (3 + 4 = 7 m); smallest when they point opposite ways (4 − 3 = 1 m). 5 m is the resultant at 90°.

  130. As the angle between two displacement vectors increases from 0° to 180°, their resultant:

    • decreases from A + B to |A − B|
    • increases from |A − B| to A + B
    • stays the same
    • increases, then decreases

    At 0° the vectors add fully; as they turn apart they cancel more and more, until at 180° only the difference is left.

  131. A vector drawn 15 mm long represents a velocity of 30 m/s. How long should a vector representing 20 m/s be?

    • 10 mm
    • 20 mm
    • 22.5 mm
    • 7.5 mm

    The scale is 15 mm ÷ 30 m/s = 0.5 mm per m/s, so 20 m/s × 0.5 mm = 10 mm.

  132. You ride a bus moving at 2.0 m/s and walk toward the front at 4.0 m/s relative to the bus. Your speed relative to the street is:

    • 6.0 m/s
    • 2.0 m/s
    • 4.0 m/s
    • 8.0 m/s

    Both velocities point the same way, so they add: 2.0 + 4.0 = 6.0 m/s relative to the street.

  133. On the same bus (2.0 m/s), you walk toward the back at 4.0 m/s relative to the bus. Your velocity relative to the street is:

    • 2.0 m/s, opposite the bus's motion
    • 6.0 m/s, in the bus's direction
    • 2.0 m/s, in the bus's direction
    • 0 m/s

    Now the velocities point opposite ways: 4.0 − 2.0 = 2.0 m/s in your walking direction, toward the back.

  134. In 1999 Hicham El Guerrouj ran a mile (1609 m) in 3 min 43.13 s. His average speed was about:

    • 7.21 m/s
    • 0.139 m/s
    • 26.8 m/s
    • 4.31 m/s

    Convert the time to seconds: 3 × 60 + 43.13 = 223.13 s. Then 1609 m ÷ 223.13 s ≈ 7.21 m/s.

  135. A motorist drives 150 km in 2.5 h and makes the return trip in 2.0 h. Her average speeds going and returning were:

    • 60 km/h and 75 km/h
    • 75 km/h and 60 km/h
    • 375 km/h and 300 km/h
    • 67.5 km/h each way

    Going: 150 ÷ 2.5 = 60 km/h. Returning: 150 ÷ 2.0 = 75 km/h.

  136. For that whole round trip (150 km each way, 2.5 h and 2.0 h), the average speed is closest to:

    • 67 km/h
    • 68 km/h
    • 60 km/h
    • 75 km/h

    Total distance ÷ total time = 300 km ÷ 4.5 h ≈ 66.7 km/h. Averaging 60 and 75 (67.5) is wrong because she spent longer at the slower speed.

  137. A hiker walks 5.0 km in the first hour and 4.4 km in the second hour. Her average speed for the first two hours is:

    • 4.7 km/h
    • 4.4 km/h
    • 9.4 km/h
    • 5.0 km/h

    (5.0 + 4.4) km ÷ 2 h = 9.4 ÷ 2 = 4.7 km/h.

  138. The same hiker (5.0 km, then 4.4 km) rests during the third hour. Her average speed for the first three hours is:

    • 3.1 km/h
    • 4.7 km/h
    • 3.5 km/h
    • 9.4 km/h

    Resting still counts as time: 9.4 km ÷ 3 h ≈ 3.1 km/h.

  139. She then walks 4.6 km in the fourth hour. Her average speed for the whole 4-hour period is:

    • 3.5 km/h
    • 4.7 km/h
    • 4.67 km/h
    • 14 km/h

    Total distance = 5.0 + 4.4 + 4.6 = 14.0 km over 4 h = 3.5 km/h. 4.67 km/h leaves out the hour of rest.

  140. The resultant of two forces acting on the same point at the same time is greatest when the angle between them is:

    • 0°
    • 45°
    • 90°
    • 180°

    At 0° the forces point the same way and add fully (A + B).

  141. Two concurrent forces of 2.0 N and 1.5 N act at right angles. The magnitude of their resultant is:

    • 2.5 N
    • 3.5 N
    • 0.5 N
    • 3.0 N

    √(2.0² + 1.5²) = √(4.00 + 2.25) = √6.25 = 2.5 N.

  142. Find the components of a helicopter's velocity of 95 km/h at 35° above the ground.

    • vx ≈ 78 km/h, vy ≈ 54 km/h
    • vx ≈ 54 km/h, vy ≈ 78 km/h
    • vx = 95 km/h, vy = 0
    • vx ≈ 67 km/h, vy ≈ 67 km/h

    vx = 95 cos 35° ≈ 95(0.819) ≈ 78 km/h; vy = 95 sin 35° ≈ 95(0.574) ≈ 54 km/h.

  143. An aircraft climbs at a steady 200. m/s at 20° above the horizontal. The horizontal and vertical components of its velocity are:

    • 68 m/s horizontal, 188 m/s vertical
    • 188 m/s horizontal, 68 m/s vertical
    • 200. m/s horizontal, 73 m/s vertical
    • 129 m/s horizontal, 153 m/s vertical

    vx = 200 cos 20° ≈ 200(0.940) ≈ 188 m/s; vy = 200 sin 20° ≈ 200(0.342) ≈ 68 m/s.

  144. A bus travels 23.0 km on a straight road 30° north of east. The east and north components of its displacement are:

    • 19.9 km east, 11.5 km north
    • 11.5 km east, 19.9 km north
    • 23.0 km east, 0 km north
    • 16.3 km east, 16.3 km north

    East = 23.0 cos 30° ≈ 19.9 km; north = 23.0 sin 30° = 11.5 km.

  145. What are the components of a vector of magnitude 1.5 m at 35° from the positive x-axis?

    • x ≈ 1.2 m, y ≈ 0.86 m
    • x ≈ 0.86 m, y ≈ 1.2 m
    • x = 1.5 m, y = 0
    • x ≈ 0.75 m, y ≈ 0.75 m

    x = 1.5 cos 35° ≈ 1.23 m; y = 1.5 sin 35° ≈ 0.86 m.

  146. A car moves at 50 km/h in a direction 60° north of east (x = east, y = north). Which component of its velocity is larger?

    • x (east): about 43 km/h vs. 25 km/h north
    • y (north): about 43 km/h vs. 25 km/h east
    • They are equal, at about 35 km/h each
    • x (east): about 50 km/h vs. 29 km/h north

    vx = 50 cos 60° = 25 km/h; vy = 50 sin 60° ≈ 43 km/h. More than 45° from the x-axis means the y-component wins.

  147. A 4 N force points east and a 3 N force points north. Their resultant is:

    • 5 N at about 37° north of east
    • 7 N east
    • 5 N at about 53° north of east
    • 1 N east

    R = √(4² + 3²) = 5 N; θ = tan⁻¹(3/4) ≈ 37° north of east. 53° measures from north instead.

  148. A velocity of 80. m/s is directed 30° above the horizontal. The magnitude of its vertical component is:

    • 40. m/s
    • 69 m/s
    • 30. m/s
    • 10. m/s

    vy = 80. sin 30° = 80.(0.50) = 40. m/s. 69 m/s is the horizontal component.

  149. In Lab 1.1, a can measures 6.6 cm across and 12.2 cm tall. Its capacity is about:

    • 417 mL
    • 1670 mL
    • 253 mL
    • 80 mL

    V = πr²h with r = 3.3 cm: π(3.3)²(12.2) ≈ 417 cm³ = 417 mL. Using the 6.6 cm diameter as r gives about 1670 mL.

  150. A student walks halfway around a 30 m × 40 m rectangular field, from one corner to the opposite corner. Her distance and displacement are:

    • 70 m and 50 m
    • 50 m and 70 m
    • 70 m and 70 m
    • 140 m and 0 m

    Distance = 30 + 40 = 70 m along the path; displacement = √(30² + 40²) = 50 m, straight across.

  151. Solving d = ½at² for t gives:

    • t = √(2d/a)
    • t = 2d/a
    • t = √(d/2a)
    • t = (2d/a)²

    Multiply both sides by 2 and divide by a to get t² = 2d/a, then take the square root.

  152. In a right triangle, sin θ equals:

    • opposite ÷ hypotenuse
    • adjacent ÷ hypotenuse
    • opposite ÷ adjacent
    • hypotenuse ÷ opposite

    SOH: sine = opposite over hypotenuse. CAH gives cosine and TOA gives tangent.

  153. A speed of 72 km/h equals:

    • 20 m/s
    • 259 m/s
    • 72 m/s
    • 2.0 m/s

    72 km/h × 1000 m/km ÷ 3600 s/h = 20 m/s (dividing by 3.6). 259 multiplies by 3.6 instead.

  154. Which pair of forces acting concurrently on an object produces the resultant of greatest magnitude?

    • 6.0 N4.0 N
    • 6.0 N4.0 N
    • 6.0 N4.0 N
    • 6.0 N4.0 N

    Forces pointing the same way add completely: 6.0 + 4.0 = 10 N. At 90° the resultant is √(36 + 16) ≈ 7.2 N, at 120° it is about 5.3 N, and opposite forces leave only 2.0 N.

  155. The diagram shows a resultant vector, R. Which pair of perpendicular component vectors, A and B, combines to form R?

    R
    • AB
    • AB
    • AB
    • AB

    R points down and to the right, so its components must point right (horizontal) and down (vertical). Placed tip-to-tail, A then B ends where R ends.

  156. Vector A (3 units east) and vector B (4 units north) are drawn tip-to-tail. Which arrow best represents their resultant, R?

    A = 3B = 4

    R runs from A's tail to B's head: 3 units east and 4 units north, so it points up and to the right at tan⁻¹(4/3) ≈ 53° above east, with length 5 units. 37° would fit 4 east and 3 north.

  157. Using vectors a (east) and b (north) shown, which arrow represents r = a − b?

    ab

    a − b = a + (−b). Reversing b makes it point south; adding it tip-to-tail to a (east) gives an arrow pointing east and south: down and to the right. Up and to the right would be a + b.

  158. The distance–time graph shows an object's motion. What is its speed?

    0123450510152025Time (s)Distance (m)
    • 5.0 m/s
    • 20. m/s
    • 0.20 m/s
    • 80. m/s

    Speed is the slope of a d–t graph: 20. m ÷ 4.0 s = 5.0 m/s. 0.20 m/s divides time by distance.

  159. Which velocity–time graph represents an object moving with a constant, nonzero velocity?

    • vt
    • vt
    • vt
    • vt

    Constant velocity means v doesn't change, so every point is at the same height: a horizontal line. A rising or falling line means the object is accelerating.

  160. Which distance–time graph represents an object at rest?

    • dt
    • dt
    • dt
    • dt

    At rest, the position never changes, so the distance stays at the same value: a horizontal line. Any sloped line means the object is moving.

  161. According to the velocity–time graph, what is the object's acceleration during the first 5.0 s?

    024568024681012Time (s)Velocity (m/s)
    • 2.0 m/s²
    • 10. m/s²
    • 0.50 m/s²
    • 50. m/s²

    Acceleration is the slope of a v–t graph: (10. m/s − 0) ÷ 5.0 s = 2.0 m/s².

  162. According to the same velocity–time graph, how far does the object travel in the full 8.0 s?

    024568024681012Time (s)Velocity (m/s)
    • 55 m
    • 80. m
    • 25 m
    • 30. m

    Distance is the area under a v–t graph: triangle ½(5.0 s)(10. m/s) = 25 m, plus rectangle (3.0 s)(10. m/s) = 30. m, for 55 m. 80. m treats the whole 8 s as 10 m/s.

  163. On the distance–time graph, during which interval is the object at rest?

    024680102030Time (s)Distance (m)ABCD
    • BC
    • AB
    • CD
    • It is never at rest

    The line is level from B to C: the distance stays at 10 m, so the object isn't moving.

  164. On the same distance–time graph, during which interval is the object moving fastest?

    024680102030Time (s)Distance (m)ABCD
    • CD
    • AB
    • BC
    • AB and CD are equally fast

    Speed is the slope. AB: 10 m ÷ 2 s = 5 m/s. CD: 20 m ÷ 2 s = 10 m/s. The steeper segment, CD, is faster.

  165. A 10.0 N force acts at 37° above the horizontal, as shown. Its horizontal and vertical components are about:

    10.0 N37°
    • 8.0 N and 6.0 N
    • 6.0 N and 8.0 N
    • 10. N and 0 N
    • 5.0 N and 5.0 N

    Horizontal = 10.0 cos 37° ≈ 8.0 N; vertical = 10.0 sin 37° ≈ 6.0 N. The component next to the angle gets the cosine.

  166. A velocity is drawn to the scale shown. The arrow is 4.0 cm long. What velocity does it represent?

    1.0 cm = 5.0 m/s4.0 cm long
    • 20. m/s
    • 4.0 m/s
    • 1.25 m/s
    • 9.0 m/s

    Each centimeter stands for 5.0 m/s, so 4.0 cm × 5.0 m/s per cm = 20. m/s.

  167. The graph shows speed versus time for objects A and B. Compared with the acceleration of B, the acceleration of A is:

    012345605101520Time (s)Speed (m/s)AB
    • three times as great
    • one-third as great
    • the same
    • nine times as great

    Acceleration is the slope. A: 15 ÷ 5 = 3 m/s². B: 5 ÷ 5 = 1 m/s². A's is three times B's.

  168. The position–time graph shows an object's motion. Which statement describes it?

    012345670102030Time (s)Position (m)
    • It speeds up as it moves away from the starting point
    • It stays at rest at a position of 30. m
    • It moves away from the start at 30. m/s
    • It moves back toward the starting point at 5.0 m/s

    The line slopes down, so the position is decreasing: the object is heading back toward 0. Its speed is the slope's size: 30. m ÷ 6.0 s = 5.0 m/s, constant because the line is straight.

  169. The diagram shows a resultant force R. Which vector is the equilibrant of R?

    R = 10. N
    • 10. N
    • 10. N
    • 5.0 N
    • 10. N

    The equilibrant cancels the resultant, so it has the same magnitude (10. N) and points the opposite way (west). Together they add to zero.

  170. Vectors A and B start at the same point, and the parallelogram is completed with dashed lines. Which line represents the resultant A + B?

    ABP
    • The diagonal joining the tip of A to the tip of B
    • The diagonal from the shared starting point to corner P
    • The dashed side parallel to A, ending at corner P
    • The dashed side parallel to B, starting at the tip of A

    In the parallelogram method the resultant is the diagonal that starts where both vectors start and ends at the far corner, P. The other diagonal (tip of A to tip of B) is B − A.

  171. A student walks from S to F along the streets shown (each square is one block). What is the magnitude of her displacement?

    34SF
    • 5 blocks
    • 7 blocks
    • 1 block
    • 12 blocks

    Displacement is the straight line from start to finish: √(3² + 4²) = √25 = 5 blocks.

  172. For the same walk from S to F, what distance did she travel?

    34SF
    • 7 blocks
    • 5 blocks
    • 12 blocks
    • 1 block

    Distance is the length of the path actually walked: 3 blocks + 4 blocks = 7 blocks. It is larger than the 5-block displacement.

  173. A student walks 10 m north and then 30 m south, as shown. What is her displacement?

    +10 m0 m-10 m-20 m10 m30 mstart
    • 20 m south
    • 40 m south
    • 20 m north
    • 40 m

    Take north as +: +10 m − 30 m = −20 m, so the displacement is 20 m south. 40 m is the distance walked.

  174. Each grid square is 1 m on a side. What is the magnitude of displacement vector d?

    d
    • 10 m
    • 14 m
    • 8 m
    • 48 m

    d has components 6 m (right) and 8 m (up): √(6² + 8²) = √100 = 10 m. 14 m just adds the components.

  175. According to the velocity–time graph, how far does the object travel in 6.0 s?

    012345670246810Time (s)Velocity (m/s)
    • 48 m
    • 1.3 m
    • 14 m
    • 8.0 m

    The area under a v–t graph is distance: a rectangle 8.0 m/s tall and 6.0 s wide = 48 m.

  176. A car's velocity–time graph is shown. What is its acceleration?

    0123450510152025Time (s)Velocity (m/s)
    • −5.0 m/s²
    • 5.0 m/s²
    • −20. m/s²
    • −0.20 m/s²

    Slope = (0 − 20. m/s) ÷ 4.0 s = −5.0 m/s². The negative sign means the car is slowing down (its velocity is positive).

  177. Which velocity–time graph shows an object slowing down uniformly until it stops?

    • vt
    • vt
    • vt
    • vt

    Uniform slowing means the velocity drops by the same amount each second: a straight line sloping down to zero.

  178. Which velocity–time graph shows an object that speeds up uniformly and then moves at constant velocity?

    • vt
    • vt
    • vt
    • vt

    Speeding up uniformly is a rising straight line; constant velocity is a horizontal line. So the graph rises, then levels off.

  179. The position–time graph shown curves upward and gets steeper. What is happening to the object?

    0123450510152025Time (s)Position (m)
    • It is speeding up
    • It is slowing down
    • It moves at constant speed
    • It is at rest

    The slope of a position–time graph is the velocity. A slope that keeps getting steeper means the velocity is increasing: the object is speeding up.

  180. The position–time graph shows two cars, A and B. What does the point where their lines cross represent?

    01234560204060Time (s)Position (m)AB
    • Both cars are traveling at the same velocity at that time
    • Both cars have the same acceleration at that moment
    • Both cars have come to rest at that moment
    • Both cars are at the same position at the same time

    Each point on a position–time graph is a position at a time. Where the lines cross, both cars share one position at that instant. Their velocities (slopes) are different, since the lines have different steepness.

  181. According to the distance–time graph, what is the object's average speed over the whole 8.0 s?

    02468010203040Time (s)Distance (m)
    • 5.0 m/s
    • 10. m/s
    • 6.7 m/s
    • 40. m/s

    Average speed = total distance ÷ total time = 40. m ÷ 8.0 s = 5.0 m/s. The time spent resting still counts; leaving it out gives 40 ÷ 6 ≈ 6.7 m/s.

  182. The dashed line is tangent to the position–time curve at t = 4.0 s. What is the object's instantaneous velocity at 4.0 s?

    024680102030Time (s)Position (m)
    • 5.0 m/s
    • 2.5 m/s
    • 10. m/s
    • 20. m/s

    Instantaneous velocity is the slope of the tangent: (20. m − 0 m) ÷ (6.0 s − 2.0 s) = 5.0 m/s. 2.5 m/s is the average velocity from 0 to 4 s (10 m ÷ 4 s).

  183. An object's velocity–time graph is a horizontal line above the time axis (shown). Which distance–time graph matches the same motion?

    vt
    • dt
    • dt
    • dt
    • dt

    A constant positive velocity means equal distances every second, so distance grows linearly: a straight rising line. A curve would mean the velocity is changing.

  184. Two forces are drawn tip-to-tail to the scale shown. The resultant R measures 5.0 cm. What is the magnitude of the resultant force?

    1.0 cm = 10. N3.0 cm4.0 cmR
    • 50. N
    • 5.0 N
    • 70. N
    • 0.50 N

    Convert with the scale: 5.0 cm × 10. N per cm = 50. N. (Check: the 3.0 cm and 4.0 cm arrows are 30. N and 40. N, and √(30² + 40²) = 50. N.)

Unit 2: Kinematics (37)

  1. A runner moves 50 m east then 20 m west. What is the runner's displacement?

    • 70 m east
    • 30 m east
    • 30 m west
    • 70 m west

    Displacement = 50 east − 20 east(since west is −) = 50 − 20 = 30 m east. Distance would be 70 m, but displacement is 30 m east.

  2. What is the total distance traveled by the runner in the previous scenario?

    • 30 m
    • 50 m
    • 20 m
    • 70 m

    Distance is the total path length: 50 m + 20 m = 70 m, regardless of direction.

  3. Which quantity is a vector?

    • Distance
    • Speed
    • Displacement
    • Time

    Displacement has both magnitude and direction, making it a vector. Distance, speed, and time are scalars.

  4. On a position-time graph, the slope represents:

    • Acceleration
    • Velocity
    • Distance
    • Displacement

    Slope of an x-t graph = $Δx/Δt =$ velocity.

  5. On a velocity-time graph, the area under the curve represents:

    • Acceleration
    • Velocity
    • Displacement
    • Force

    Area under a v-t graph = displacement (since $Δx = v·t$ for each small interval).

  6. An object has a positive velocity and a negative acceleration. The object is:

    • Speeding up
    • Slowing down
    • Moving at constant speed
    • Not moving

    When velocity and acceleration have opposite signs, the object is slowing down (decelerating).

  7. A car accelerates from rest at 4 m/s² for 5 seconds. What is its final velocity?

    • 9 m/s
    • 20 m/s
    • 0.8 m/s
    • 25 m/s

    vf = vi + at = 0 + (4)(5) = 20 m/s.

  8. A ball is thrown straight up. At the very top of its path, its acceleration is:

    • Zero
    • Equal to g, directed downward
    • Equal to g, directed upward
    • Undefined

    Gravity acts continuously; acceleration is always −g (downward) even at the peak where velocity is momentarily zero.

  9. An object starts from rest and accelerates uniformly at 3 m/s² for 4 seconds. How far does it travel?

    • 12 m
    • 24 m
    • 6 m
    • 48 m

    $Δx = vi·t +$ ½at² = 0 + ½(3)(4²) = ½(3)(16) = 24 m.

  10. A projectile is launched horizontally from a cliff. Compared to an object simply dropped from the same height at the same time, the projectile:

    • Hits the ground first
    • Hits the ground later
    • Hits the ground at the same time
    • Never hits the ground

    Horizontal and vertical motions are independent; both objects have the same initial vertical velocity (0) and the same vertical acceleration (g), so they fall for the same amount of time.

  11. A ball is launched at 20 m/s at an angle of 30° above the horizontal. What is its initial vertical velocity component?

    • 20 m/s
    • 17.3 m/s
    • 10 m/s
    • 0 m/s

    vyi = $v \sinθ = 20$ sin30° = 20(0.5) = 10 m/s.

  12. For maximum range on level ground (no air resistance), a projectile should be launched at an angle of:

    • 30°
    • 45°
    • 60°
    • 90°

    Range is maximized at 45° for equal launch and landing heights, ignoring air resistance.

  13. Which of the following can be true for an object with zero velocity?

    • It must have zero acceleration
    • It cannot have nonzero acceleration
    • It can have nonzero acceleration
    • It must be at its starting position

    An object can be momentarily at rest while still accelerating, such as a ball at the top of its toss, or a car just starting to move.

  14. A car's velocity-time graph is a horizontal line above the time axis. This means the car has:

    • Constant positive acceleration
    • Zero velocity
    • Constant nonzero velocity
    • Increasing velocity

    A horizontal (flat) line on a v-t graph means velocity isn't changing — constant velocity, zero acceleration.

  15. An object's average speed over a trip is 20 m/s. Its average velocity for the same trip:

    • Could be less than 20 m/s in magnitude
    • Must also equal 20 m/s in magnitude
    • Must be greater than 20 m/s in magnitude
    • Must be negative along the path taken

    Average speed = total distance/time; average velocity = displacement/time. Since distance ≥ |displacement|, average speed ≥ |average velocity|.

  16. A ball is dropped from rest and falls for 2.0 s. How far does it fall (use g=9.8 m/s²)?

    • 9.8 m
    • 19.6 m
    • 4.9 m
    • 39.2 m

    $Δx =$ ½gt² = ½(9.8)(2.0²) = ½(9.8)(4) = 19.6 m.

  17. Which kinematic equation should be used if you know vi, a, and $Δx$, and want to find vf without knowing t?

    • vf = vi + at
    • $Δx =$ vi t + ½at²
    • vf² = vi² $+ 2aΔx$
    • $Δx =$ ½(vi+vf)t

    vf² = vi² $+ 2aΔx$ is the only kinematic equation that omits time.

  18. A car speeds up from 12 m/s to 30 m/s in 6.0 s. What is its acceleration?

    • $3.0 m/s^2$
    • $5.0 m/s^2$
    • $0.33 m/s^2$
    • $18 m/s^2$

    $a = (30 - 12)/6.0 = 3.0 m/s^2$.

  19. Starting from rest, an object accelerates $at 4.0 m/s^2$ for 5.0 s. How far does it travel?

    • 20 m
    • 50 m
    • 100 m
    • 10 m

    dx = $0.5(4.0)(5.0)^2 = 50 m$.

  20. A ball is dropped from rest. How fast is it moving after $3.0 s (g = 9.8 m/s^2)?$

    • 3.3 m/s
    • 19.6 m/s
    • 29.4 m/s
    • 44.1 m/s

    v = gt = 9.8 x 3.0 = 29.4 m/s.

  21. A runner jogs 300 m east, then 100 m west. What is the magnitude of the displacement?

    • 400 m
    • 200 m
    • 100 m
    • 0 m

    300 east - 100 west = 200 m east; distance would be 400 m.

  22. On a velocity-time graph, what does the slope of the line represent?

    • Displacement
    • Acceleration
    • Distance
    • Position

    The slope of a v-t graph is the rate of change of velocity, i.e. acceleration.

  23. A stone is thrown straight up at 20 m/s. Approximately how long until it returns to the thrower's hand $(g = 10 m/s^2)?$

    • 2 s
    • 4 s
    • 1 s
    • 20 s

    Time up = 20/10 = 2 s; total flight is symmetric, so about 4 s.

  24. A projectile is launched horizontally at 15 m/s from a 20 m cliff $(g = 10 m/s^2)$. How long is it in the air?

    • 1.3 s
    • 2.0 s
    • 4.0 s
    • 3.0 s

    $20 = 0.5(10)t^2$ gives $t^2 = 4, t = 2.0 s$; horizontal speed does not affect fall time.

  25. For that same projectile (15 m/s horizontal, 2.0 s aloft), how far from the base of the cliff does it land?

    • 15 m
    • 30 m
    • 20 m
    • 45 m

    Range = $v_x t = 15 x 2.0 = 30 m$.

  26. A car moving at 25 m/s brakes to a stop over 50 m. What is its acceleration?

    • $-6.25 m/s^2$
    • $-12.5 m/s^2$
    • $-2.5 m/s^2$
    • $-0.5 m/s^2$

    $0 = 25^2 + 2a(50)$ gives $a = -625/100 = -6.25 m/s^2$.

  27. At the highest point of its path, a ball thrown straight up has:

    • zero velocity and zero acceleration
    • zero velocity and downward acceleration g
    • maximum velocity and zero acceleration
    • downward velocity and upward acceleration

    Vertical velocity is momentarily zero, but gravity still acts, so acceleration is g downward.

  28. Which launch angle gives the maximum range for a projectile on level ground (no air resistance)?

    • 30 degrees
    • 45 degrees
    • 60 degrees
    • 90 degrees

    Range is proportional to sin(2theta), which is largest at theta = 45 degrees.

  29. Distance traveled is always:

    • less than or equal to the magnitude of displacement
    • equal to the magnitude of displacement in every case
    • greater than or equal to the magnitude of displacement
    • a vector quantity pointing from start to finish

    Distance is the total path length and can never be less than the straight-line displacement magnitude.

  30. An astronaut drops a hammer from 2.0 m above the Moon's surface, where g = 1.62 m/s². How long does it take to reach the surface?

    • 1.6 s
    • 1.2 s
    • 2.5 s
    • 0.62 s

    Starting from rest, d = ½gt², so t = √(2d ÷ g) = √(4.0 ÷ 1.62) = √2.47 ≈ 1.6 s.

  31. A coin and a feather are dropped together inside a tube with the air pumped out. They:

    • hit the bottom at the same time
    • fall so the coin lands first
    • fall so the feather lands first
    • float, because there is no gravity in a vacuum

    Without air resistance, gravity gives every object the same acceleration, so they fall together. Gravity still acts in a vacuum.

  32. Who taught that heavier objects fall faster, in proportion to their weight?

    • Aristotle
    • Galileo
    • Newton
    • Einstein

    Aristotle held that view. Galileo later showed that, apart from air resistance, all bodies fall with the same acceleration.

  33. An object is dropped from rest. Ignoring air resistance (g = 9.8 m/s²), its speed after 2.0 s is about:

    • 20 m/s
    • 9.8 m/s
    • 4.9 m/s
    • 39 m/s

    v = gt = (9.8)(2.0) = 19.6 ≈ 20 m/s. Each second of free fall adds 9.8 m/s.

  34. In a reaction-time test, a ruler falls 0.20 m before a student catches it. The student's reaction time is about (g = 9.8 m/s²):

    • 0.20 s
    • 0.041 s
    • 0.45 s
    • 2.0 s

    From rest, t = √(2d/g) = √(2 × 0.20 ÷ 9.8) = √0.041 ≈ 0.20 s. 0.041 forgets the square root.

  35. A ball is thrown straight up. At the very top of its flight, its velocity and acceleration are:

    • v = 0 and a = 9.8 m/s² downward
    • v = 0 and a = 0
    • v = 9.8 m/s and a = 0
    • v = 0 and a = 9.8 m/s² upward

    The ball stops for an instant, but gravity never switches off, so the acceleration is still g downward.

  36. A rock is dropped from rest off a 20. m ledge. Ignoring air resistance (g = 9.8 m/s²), it reaches the ground in about:

    • 2.0 s
    • 4.1 s
    • 1.0 s
    • 0.49 s

    t = √(2d/g) = √(40. ÷ 9.8) = √4.1 ≈ 2.0 s. 4.1 s forgets the square root.

  37. A ball is thrown straight up at 15 m/s (g = 9.8 m/s²). How long does it take to reach the top?

    • 1.5 s
    • 3.1 s
    • 0.65 s
    • 15 s

    At the top v = 0, so 0 = 15 − 9.8t and t = 15 ÷ 9.8 ≈ 1.5 s. 3.1 s is the whole trip up and back down.

Unit 3: Newton's Laws & Dynamics (29)

  1. An object moves at constant velocity. What can you conclude about the net force acting on it?

    • It must be zero
    • It must equal the object's weight
    • It must be increasing
    • It cannot be determined

    Constant velocity means zero acceleration, so by Newton's Second Law, the net force must be zero.

  2. A 10 kg object experiences a net force of 25 N. What is its acceleration?

    • 250 m/s²
    • 2.5 m/s²
    • 0.4 m/s²
    • 15 m/s²

    a = Fnet/m = 25/10 = 2.5 m/s².

  3. Which pair of forces is an example of a Newton's Third Law force pair?

    • Weight and normal force on the same object
    • The force of a hammer on a nail, and the nail on the hammer
    • Tension in a rope and gravity on a hanging object
    • Friction and applied force on the same object

    Action-reaction pairs act on two DIFFERENT objects — the hammer pushes the nail, the nail pushes back on the hammer with equal and opposite force.

  4. A book rests on a table. The normal force on the book is equal in magnitude to the book's weight. This is an example of:

    • A Newton's Third Law action–reaction pair acting on different objects
    • A centripetal force pair directed toward the center of the table
    • An impulse–momentum pair acting over a short time interval
    • Two forces on the same object that happen to balance (equilibrium)

    Weight and normal force act on the SAME object (the book) and happen to balance since the book isn't accelerating vertically — NOT a third-law pair, which would require different objects.

  5. What is the weight of a 5 kg mass on Earth (g = 9.8 m/s²)?

    • 5 N
    • 49 N
    • 0.51 N
    • 5 kg

    Fg = mg = (5)(9.8) = 49 N.

  6. A box on a horizontal surface has a mass of 8 kg and a coefficient of kinetic friction of 0.3. What is the kinetic friction force (g=9.8 m/s²)?

    • 2.4 N
    • 23.5 N
    • 78.4 N
    • 0.24 N

    FN = mg = (8)(9.8) = 78.4 N. fk = $μkFN = (0.3)(78.4) = 23.5 N$.

  7. Which statement about mass and weight is correct?

    • Mass and weight are the same thing measured in different units
    • Mass is constant everywhere; weight depends on local gravity
    • Weight is constant everywhere; mass depends on local gravity
    • Both mass and weight change with location

    Mass (kg) measures the amount of matter and doesn't change; weight (N) = mg depends on the local gravitational field.

  8. A rope holds a 12 kg mass stationary, hanging vertically. What is the tension in the rope (g=9.8 m/s²)?

    • 12 N
    • 1.2 N
    • 117.6 N
    • 0 N

    Since the mass is in equilibrium, tension equals weight: T = mg = (12)(9.8) = 117.6 N.

  9. An astronaut has a mass of 70 kg on Earth. On the Moon, where gravity is about 1/6 of Earth's, the astronaut's mass is:

    • 70 kg still
    • About 11.7 kg
    • 420 kg
    • Zero, since there's no gravity in space

    Mass does not change with location — only weight changes. The astronaut's mass remains 70 kg on the Moon.

  10. Which of the following increases the maximum static friction force between two surfaces?

    • Increasing the contact area
    • Increasing the normal force
    • Decreasing the normal force
    • Increasing the object's velocity

    Maximum static friction $fs,max = μsFN$ depends on the normal force (and $μ$), not on contact area or velocity.

  11. A block sits on a frictionless incline at angle $θ$. The component of gravity acting parallel to the incline surface (down the slope) is:

    • mg
    • mg $\cosθ$
    • mg $\sinθ$
    • mg $\tanθ$

    The component of weight parallel to the incline is mg $\sinθ$; the component perpendicular (balanced by normal force) is mg $\cosθ$.

  12. Two blocks are connected by a rope over a frictionless pulley. Which law explains why the same magnitude of acceleration applies to both blocks?

    • Conservation of momentum, since the blocks form a closed system
    • Newton's First Law, since each block keeps its state of motion
    • The rope is inextensible, constraining both blocks to move together
    • Newton's Third Law, since the blocks exert equal and opposite forces

    Because the rope doesn't stretch, both blocks must have the same speed and acceleration magnitude at all times — a constraint, not directly a Newton's law statement.

  13. Which of these is NOT one of Newton's Laws of Motion?

    • An object in motion stays in motion unless acted on by a net force
    • Force equals mass times acceleration
    • For every action there is an equal and opposite reaction
    • Energy cannot be created or destroyed

    'Energy cannot be created or destroyed' is the law of conservation of energy, not one of Newton's three laws of motion.

  14. A 1000 kg car accelerates at 2 m/s². What net force is required?

    • 500 N
    • 2000 N
    • 1000 N
    • 4000 N

    Fnet = ma = (1000)(2) = 2000 N.

  15. When you push against a wall, the wall pushes back on you with equal force. This is best explained by:

    • Newton's First Law
    • Newton's Second Law
    • Newton's Third Law
    • Conservation of energy

    Newton's Third Law: for every action force, there's an equal and opposite reaction force.

  16. An object is in equilibrium on an incline, held by a rope parallel to the incline. The tension in the rope is:

    • mg
    • mg $\cosθ$
    • mg $\sinθ$
    • zero

    For equilibrium along the incline, tension must balance the component of gravity along the incline: T = mg $\sinθ$.

  17. Which best describes inertia?

    • The force that keeps objects moving
    • A property that measures the amount of gravitational force on an object
    • The tendency of an object to resist a change in its state of motion
    • The rate of change of momentum

    Inertia is the tendency of matter to resist changes in velocity (both magnitude and direction); mass is the quantitative measure of inertia.

  18. A 5.0 kg block is pushed with a net force of 20 N. What is its acceleration?

    • $4.0 m/s^2$
    • $100 m/s^2$
    • $0.25 m/s^2$
    • $15 m/s^2$

    $a = F_{net}/m = 20/5.0 = 4.0 m/s^2$.

  19. What is the weight of a 12 kg object near Earth's surface $(g = 9.8 m/s^2)?$

    • 12 N
    • 1.2 N
    • 118 N
    • 98 N

    W = mg = 12 x 9.8 = 117.6 N, about 118 N.

  20. An object moves at constant velocity. The net force on it is:

    • in the direction of motion
    • zero
    • opposite to the motion
    • equal to its weight

    Constant velocity means zero acceleration, so the net force is zero (equilibrium).

  21. A book rests on a table. The reaction partner (Newton's third law) to Earth's gravitational pull on the book is:

    • the table's normal force on the book
    • the book's gravitational pull on Earth
    • the book's push down on the table
    • Earth's pull on the table

    The action-reaction partner of Earth pulling the book is the book pulling Earth with equal, opposite force.

  22. A 2.0 kg box on a level floor has $mu_k = 0.30$. What is the kinetic friction force while it slides $(g = 10 m/s^2)?$

    • 6.0 N
    • 0.60 N
    • 20 N
    • 0.15 N

    $f_k = mu_k N = 0.30 x (2.0 x 10) = 6.0 N$.

  23. On a frictionless incline at 30 degrees, the acceleration of a sliding block is $(g = 10 m/s^2)$:

    • $10 m/s^2$
    • $5.0 m/s^2$
    • $8.7 m/s^2$
    • zero

    $a = g \sin(30) = 10 x 0.5 = 5.0 m/s^2$.

  24. Doubling the net force on an object while keeping its mass the same will:

    • double the acceleration
    • halve the acceleration
    • not change the acceleration
    • double the mass

    a is proportional to $F_{net}$, so doubling the force doubles the acceleration.

  25. The normal force on a book lying on a horizontal table (no other vertical forces) equals:

    • zero
    • the book's weight
    • twice the book's weight
    • the applied horizontal force

    With no vertical acceleration, the upward normal force balances the downward weight.

  26. Which statement about mass and weight is correct?

    • Weight changes with location; mass does not
    • Mass changes with location; weight does not
    • Both are measured in newtons at any location
    • They are the same quantity, measured in kilograms

    Mass is an amount of matter (constant); weight = mg depends on local g.

  27. A 10 N horizontal force is applied to a 4.0 kg box, and friction is 2.0 N. The acceleration is:

    • $2.0 m/s^2$
    • $2.5 m/s^2$
    • $3.0 m/s^2$
    • $0.8 m/s^2$

    $F_{net} = 10 - 2.0 = 8.0 N$; $a = 8.0/4.0 = 2.0 m/s^2$.

  28. Static friction acting on a stationary crate that no one is pushing is:

    • equal to $mu_s N$
    • zero
    • equal to the crate's weight
    • directed downward

    Static friction only acts to oppose an applied force or tendency to slide; with no push it is zero.

  29. A tension of 40 N supports a hanging mass at rest. The mass is $(g = 10 m/s^2)$:

    • 4.0 kg
    • 400 kg
    • 0.4 kg
    • 40 kg

    T = mg, so m = 40/10 = 4.0 kg.

Unit 4: Circular Motion & Gravitation (28)

  1. A car travels in a circle at constant speed. Which statement is true?

    • The car has zero acceleration because its speed is not changing
    • The car has acceleration directed toward the center of the circle
    • The car has acceleration directed away from the center of the circle
    • The car has acceleration directed along its direction of motion

    Even at constant speed, direction is continuously changing, producing centripetal acceleration directed toward the center.

  2. A 1200 kg car rounds a curve of radius 50 m at 10 m/s. What centripetal force is required?

    • 240 N
    • 2400 N
    • 120 N
    • 24000 N

    Fc = mv²/r = (1200)(10²)/50 = (1200)(100)/50 = 2400 N.

  3. What force provides the centripetal force for a satellite orbiting Earth?

    • Tension
    • Friction
    • Gravity
    • Normal force

    Gravity between the satellite and Earth supplies the necessary centripetal force for circular orbit.

  4. If the distance between two masses is tripled, the gravitational force between them:

    • Triples
    • Is 1/3 as strong
    • Is 1/9 as strong
    • Is 9 times as strong

    Gravity follows an inverse-square law: Fg ∝ 1/r². Tripling r reduces force to 1/3² = 1/9.

  5. Two masses are doubled (both m1 and m2 double) while distance stays the same. The gravitational force between them becomes:

    • 2 times as strong
    • 4 times as strong
    • Half as strong
    • Unchanged

    Fg ∝ m1m2, so doubling both masses multiplies the force by 2×2=4.

  6. A ball on a string is swung in a horizontal circle. What provides the centripetal force?

    • Gravity
    • Air resistance
    • Tension in the string
    • The ball's inertia

    Tension in the string pulls the ball toward the center, providing the centripetal force.

  7. According to Kepler's Third Law, if planet A orbits at twice the radius of planet B (same central star), planet A's orbital period compared to planet B's is:

    • 2 times as long
    • 4 times as long
    • $2√2$ times as long
    • Half as long

    T² ∝ r³, so TA²/TB² = (2)³ = 8 → $TA/TB = √8 = 2√2$ ≈ 2.83.

  8. An object moves in a circle at constant speed. Which of the following is constant?

    • Velocity
    • Centripetal acceleration direction only relative to a fixed axis
    • Speed
    • Position

    Speed (magnitude of velocity) stays constant in uniform circular motion, but velocity direction and acceleration direction continuously change (though acceleration magnitude is constant, its direction always points to the center, which itself is a changing direction in space).

  9. Universal gravitation constant G has units of:

    • N/kg
    • N·m²/kg²
    • m/s²
    • kg·m/s

    From Fg = Gm1m2/r², G must have units of N·m²/kg² so the equation balances dimensionally.

  10. A satellite orbits at a larger radius than another satellite around the same planet. Compared to the closer satellite, the farther one has:

    • A greater orbital speed
    • A smaller orbital speed
    • The same orbital speed
    • An orbital speed independent of radius

    Orbital speed $v=√(Gm/r)$ decreases as r increases — farther satellites move slower.

  11. Why is there no real outward force acting on a passenger in a car turning a corner (in an inertial frame)?

    • Because the passenger's inertia makes them tend to continue in a straight line, while the car curves under them
    • Because friction between the seat and the passenger cancels every other force acting on them
    • Because gravity pulls the passenger sideways toward the outside of the curve as the car turns
    • Because the passenger is pushed outward by a real centrifugal force that the car's motion creates

    In an inertial reference frame, only a real inward (centripetal) net force acts; the sensation of being 'thrown outward' is simply the passenger's inertia resisting the change in direction.

  12. A 500 kg satellite and a 1000 kg satellite orbit Earth at the same radius. Which has the greater orbital speed?

    • The 500 kg satellite
    • The 1000 kg satellite
    • Both have the same orbital speed
    • Cannot be determined

    Orbital speed $v=√(GMEarth/r)$ does not depend on the orbiting satellite's own mass — both have the same speed at the same radius.

  13. Kepler's Second Law states that a planet sweeps out equal areas in equal times. This implies the planet moves:

    • Fastest when farthest from the Sun
    • Fastest when closest to the Sun
    • At constant speed throughout its orbit
    • Only in a perfect circle

    To sweep equal areas in equal times while closer to the Sun (a shorter radius arm), the planet must move faster near perihelion.

  14. What is the centripetal acceleration of an object moving at 4 m/s in a circle of radius 2 m?

    • 2 m/s²
    • 8 m/s²
    • 16 m/s²
    • 4 m/s²

    ac = v²/r = (4²)/2 = 16/2 = 8 m/s².

  15. A planet's orbit is best described by Kepler's First Law as:

    • A circle with the Sun at its exact center
    • A parabola with the Sun at its vertex
    • An ellipse with the Sun at one focus
    • An ellipse with the Sun at its exact center

    Kepler's First Law: planetary orbits are ellipses with the Sun at one of the two foci.

  16. If Earth's radius suddenly doubled with mass unchanged, the gravitational force on an object at the new surface would be:

    • The same
    • Twice as strong
    • Half as strong
    • One-fourth as strong

    Fg ∝ 1/r²; doubling the radius (distance to center) reduces surface gravity to 1/4.

  17. A 0.50 kg ball on a 1.0 m string moves in a circle at 4.0 m/s. The centripetal acceleration is:

    • $16 m/s^2$
    • $8.0 m/s^2$
    • $4.0 m/s^2$
    • $2.0 m/s^2$

    $a_c = v^2/r = 16/1.0 = 16 m/s^2$.

  18. For that ball (0.50 kg, $a_c = 16 m/s^2$), the required centripetal force is:

    • 8.0 N
    • 16 N
    • 4.0 N
    • 32 N

    $F = m a_c = 0.50 x 16 = 8.0 N$.

  19. Centripetal acceleration always points:

    • tangent to the circle
    • toward the center of the circle
    • away from the center
    • in the direction of motion

    It is the acceleration that continually turns the velocity toward the center.

  20. If the distance between two masses triples, the gravitational force between them becomes:

    • 3 times as large
    • 1/3 as large
    • 1/9 as large
    • 9 times as large

    Gravity follows an inverse-square law: force is proportional to $1/r^2$.

  21. An astronaut feels weightless in an orbiting station because:

    • there is essentially no gravity at that height above Earth
    • the station's spin cancels out gravity at the center
    • the station and astronaut are in free fall together
    • the air inside the cabin supports the astronaut's weight

    Both fall toward Earth with the same acceleration, so there is no normal force between them.

  22. The force that keeps a car moving around a flat, level curve is:

    • the normal force from the road on the car
    • static friction between tires and road
    • the car's weight pulling it toward the ground
    • the engine's forward thrust on the wheels

    Static friction points toward the center of the curve and supplies the centripetal force.

  23. Surface gravitational field strength is g = 9.8 N/kg. This is numerically equal to:

    • $9.8 m/s^2$
    • 9.8 J
    • 9.8 N
    • 9.8 kg

    N/kg and $m/s^2$ are equivalent units; both describe the free-fall acceleration.

  24. For satellites orbiting Earth, a larger orbital radius corresponds to:

    • a faster orbital speed
    • a slower orbital speed
    • the same orbital speed
    • zero orbital speed

    $v = \sqrt{GM/r}$, so speed decreases as r increases.

  25. At the top of a vertical circular loop, the forces providing the centripetal force on a car are:

    • gravity minus the normal force (they point in opposite directions)
    • gravity plus the normal force (both point down toward the center)
    • the normal force alone (gravity is balanced out at the top)
    • friction between the tires and the track (pointing toward the center)

    At the top, both weight and the track's normal force point downward, toward the center.

  26. Doubling the speed of an object in uniform circular motion (same radius) changes the centripetal force by a factor of:

    • 2
    • 4
    • 1/2
    • 1

    $F_c$ is proportional to $v^2$, so doubling v quadruples the force.

  27. Weight differs on the Moon compared with Earth because:

    • the object's mass is different on the Moon
    • the gravitational field strength g is different
    • the object's volume shrinks in a weaker field
    • the Moon's distance from the Sun is different

    Weight = mg, and the Moon's surface g is about 1/6 of Earth's.

  28. The period of an object in uniform circular motion is the time to:

    • reach maximum speed
    • complete one full revolution
    • travel one radius
    • reach the center

    Period T is the time for one complete trip around the circle.

Unit 5: Rotational Motion & Torque (12)

  1. One complete revolution is equal to how many radians?

    • pi
    • 2 pi
    • pi/2
    • 360

    One full circle is 2 pi radians (equivalently 360 degrees).

  2. Torque is calculated as:

    • force times the perpendicular lever arm
    • force divided by the lever arm distance
    • mass times the linear acceleration
    • force times the time it acts

    Torque = r F sin(theta), i.e. force times the perpendicular distance to the axis.

  3. A force applied directly through the axis of rotation produces:

    • maximum torque
    • zero torque
    • negative torque
    • a linear acceleration only

    The lever arm is zero, so the torque is zero regardless of the force's size.

  4. An object is in rotational equilibrium when:

    • it is not moving
    • the net torque on it is zero
    • its angular velocity is maximum
    • all forces point toward the axis

    Rotational equilibrium requires the sum of torques (and forces) to be zero.

  5. Moment of inertia depends on:

    • the mass and how it is distributed relative to the axis
    • the total mass, regardless of where it is located
    • the angular velocity the object is spinning with
    • the torque that is applied about the axis

    I depends on each mass element and the square of its distance from the axis.

  6. A skater spinning with arms out pulls them in. Her angular speed:

    • decreases
    • increases
    • stays the same
    • drops to zero

    Pulling in reduces I; with angular momentum conserved, omega increases.

  7. The rotational form of Newton's second law is:

    • F = ma
    • net torque = I alpha
    • p = m v
    • W = F d

    Net torque equals moment of inertia times angular acceleration.

  8. A door handle is placed far from the hinges because this:

    • lowers the mass the hinges must support
    • decreases the door's moment of inertia
    • adds strength to the hinges themselves
    • increases the torque for a given push

    A longer lever arm produces more torque from the same force.

  9. Angular momentum of a rotating body is:

    • I omega
    • $0.5 I omega^2$
    • m v
    • r F sin(theta)

    L = I omega; it is conserved when no external torque acts.

  10. A solid sphere and a hoop of equal mass and radius roll down the same ramp from rest. The one reaching the bottom first is:

    • the hoop
    • the solid sphere
    • they tie
    • it depends on the ramp color

    The sphere has a smaller $I/(MR^2)$, so more of its energy goes into translation; it wins.

  11. A wheel rotating at 3.0 rad/s speeds up uniformly to 9.0 rad/s in 2.0 s. Its angular acceleration is:

    • $3.0 rad/s^2$
    • $6.0 rad/s^2$
    • $12 rad/s^2$
    • $1.5 rad/s^2$

    alpha = $(9.0 - 3.0)/2.0 = 3.0 rad/s^2$.

  12. All points on a rigid spinning disk share the same:

    • linear speed
    • angular velocity
    • distance from the axis
    • centripetal acceleration

    Every point turns through the same angle in the same time, so omega is common; linear speed grows with radius.

Unit 6: Momentum & Energy (31)

  1. A 2 kg object moves at 5 m/s. What is its momentum?

    • 2.5 kg·m/s
    • 10 kg·m/s
    • 7 kg·m/s
    • 0.4 kg·m/s

    p = mv = (2)(5) = 10 kg·m/s.

  2. A 3 kg object experiences a net force of 6 N for 2 seconds. What is the impulse delivered?

    • 12 N·s
    • 3 N·s
    • 18 N·s
    • 1 N·s

    $J = FΔt = (6)(2) = 12 N·s$.

  3. Airbags reduce injury in car crashes primarily by:

    • Increasing the time over which momentum changes, reducing the force
    • Reducing the passenger's change in momentum so less impulse is needed
    • Increasing the force on the passenger while shortening the stopping time
    • Eliminating the passenger's momentum before the collision begins

    For a fixed impulse $(Δp)$, extending the time of impact reduces the average force $(J=FΔt)$, which is how airbags reduce injury.

  4. Two objects of equal mass moving in opposite directions with equal speed collide and stick together. What is their final velocity?

    • Equal to the original speed
    • Zero
    • Double the original speed
    • Cannot be determined

    Total initial momentum = mv + m(−v) = 0, so by conservation of momentum, final momentum must also be zero, meaning they stop.

  5. Which type of collision conserves both momentum AND kinetic energy?

    • Perfectly inelastic collision
    • Elastic collision
    • Any collision involving a large object
    • Explosions only

    Elastic collisions conserve both momentum and total kinetic energy; inelastic collisions conserve only momentum.

  6. A 4 kg cart moving at 3 m/s collides and sticks with a stationary 2 kg cart. What is their common final velocity?

    • 1 m/s
    • 2 m/s
    • 3 m/s
    • 6 m/s

    Conservation of momentum: (4)(3) + (2)(0) = (4+2)v → 12 = 6v → v = 2 m/s.

  7. How much work is done by a 10 N force pushing an object 5 m in the direction of the force?

    • 2 J
    • 15 J
    • 50 J
    • 0.5 J

    W = Fd $\cosθ =$ (10)(5)(cos0°) = 50 J.

  8. A person carries a heavy bag horizontally at constant velocity across a room. How much work does the person do on the bag (ignoring the initial lift)?

    • A large positive amount
    • A large negative amount
    • Zero
    • Depends on the bag's weight only

    The applied force (upward, to support the bag) is perpendicular to the horizontal displacement, so W = Fd cos90° = 0.

  9. What is the kinetic energy of a 2 kg object moving at 6 m/s?

    • 12 J
    • 24 J
    • 36 J
    • 72 J

    KE = ½mv² = ½(2)(6²) = ½(2)(36) = 36 J.

  10. If the speed of an object doubles, its kinetic energy:

    • Doubles
    • Triples
    • Quadruples
    • Stays the same

    KE ∝ v², so doubling speed quadruples kinetic energy.

  11. A 5 kg object is raised 3 m above the ground. What is its gravitational potential energy relative to the ground (g=9.8 m/s²)?

    • 15 J
    • 147 J
    • 1.67 J
    • 24.5 J

    PE = mgh = (5)(9.8)(3) = 147 J.

  12. A roller coaster car at the top of a hill has PE = 5000 J and KE = 0 (ignoring friction). What is its KE at the bottom of the hill if PE there is 0?

    • 0 J
    • 2500 J
    • 5000 J
    • 10000 J

    By conservation of mechanical energy, total energy stays constant: KEf = PEi − PEf + KEi = 5000 − 0 + 0 = 5000 J.

  13. In a real roller coaster with friction, the total mechanical energy at the bottom of a hill compared to the top is:

    • The same, because energy is conserved even with friction present
    • Greater, because gravity adds extra kinetic energy on the way down
    • Less, because some energy converts to heat/sound
    • Zero, because all of the energy is converted to heat by the bottom

    Friction is a non-conservative force that removes mechanical energy from the system, converting it to heat and sound — total energy (including heat) is still conserved, but mechanical energy decreases.

  14. Which quantity is conserved in ALL collisions, whether elastic or inelastic (assuming an isolated system)?

    • Kinetic energy
    • Momentum
    • Both momentum and kinetic energy
    • Neither

    Momentum is always conserved in an isolated system's collisions; kinetic energy is only conserved in elastic collisions.

  15. A spring with spring constant k=200 N/m is compressed 0.1 m. What is the elastic potential energy stored?

    • 20 J
    • 2 J
    • 1 J
    • 200 J

    PEspring = ½kx² = ½(200)(0.1²) = ½(200)(0.01) = 1 J.

  16. How much work is done lifting a 20 N object 3.0 m straight up at constant speed?

    • 6.7 J
    • 60 J
    • 23 J
    • 0 J

    W = Fd = 20 x 3.0 = 60 J.

  17. A 2.0 kg object moves at 4.0 m/s. Its kinetic energy is:

    • 8.0 J
    • 16 J
    • 4.0 J
    • 32 J

    KE = $0.5(2.0)(4.0)^2 = 16 J$.

  18. A force perpendicular to an object's displacement does how much work?

    • Maximum work
    • Zero work
    • Negative work
    • Half the force times distance

    W = Fd cos(90) = 0.

  19. A 1.0 kg ball is dropped from 5.0 m. Its speed just before landing is ($g = 10 m/s^2$, no air resistance):

    • 10 m/s
    • 5.0 m/s
    • 50 m/s
    • 7.1 m/s

    mgh = $0.5 m v^2$ gives v = sqrt(2 x 10 x 5.0) = 10 m/s.

  20. A motor does 6000 J of work in 30 s. Its power output is:

    • 200 W
    • 180000 W
    • 30 W
    • 2 W

    P = W/t = 6000/30 = 200 W.

  21. The momentum of a 3.0 kg cart moving at 2.0 m/s is:

    • 1.5 kg*m/s
    • 6.0 kg*m/s
    • 5.0 kg*m/s
    • 12 kg*m/s

    p = mv = 3.0 x 2.0 = 6.0 kg*m/s.

  22. A 0.10 kg ball hits a wall at 8.0 m/s and bounces straight back at 6.0 m/s. The magnitude of the impulse on the ball is:

    • 0.20 N*s
    • 1.4 N*s
    • 0.14 N*s
    • 0.80 N*s

    J = m*delta v = 0.10 x (8.0 - (-6.0)) = 0.10 x 14 = 1.4 N*s.

  23. Two objects stick together after colliding. This collision is:

    • perfectly elastic
    • perfectly inelastic
    • partially elastic with KE conserved
    • impossible

    Objects moving off with a common velocity define a perfectly inelastic collision.

  24. In every collision in an isolated system, which quantity is always conserved?

    • Kinetic energy
    • Total momentum
    • Speed
    • Force

    Total momentum is conserved in all collisions; kinetic energy only in elastic ones.

  25. A car's speed doubles. Its kinetic energy becomes:

    • twice as large
    • four times as large
    • half as large
    • unchanged

    KE is proportional to $v^2$, so doubling v quadruples KE.

  26. A 500 N force pushes a crate 4.0 m across a floor. If 300 J is lost to friction, the crate's kinetic energy increases by:

    • 2000 J
    • 1700 J
    • 300 J
    • 2300 J

    $W_{net} = 500 x 4.0 - 300 = 2000 - 300 = 1700 J =$ delta KE.

  27. Airbags reduce injury in a crash mainly because they:

    • reduce the change in momentum, lowering the impulse
    • increase the collision time, lowering the peak force
    • increase the force on the passenger, shortening the stop
    • conserve the car's kinetic energy through the collision

    For a fixed change in momentum, a longer contact time means a smaller average force.

  28. A 60 kg skater at rest throws a 2.0 kg ball east at 9.0 m/s. The skater's recoil speed is about:

    • 0.30 m/s west
    • 0.30 m/s east
    • 9.0 m/s west
    • 3.0 m/s west

    0 = 2.0(9.0) + 60(v) gives v = -0.30 m/s (west).

  29. Gravitational potential energy near Earth's surface is calculated as:

    • $0.5 m v^2$
    • mgh
    • F d cos(theta)
    • $0.5 k x^2$

    $PE_{gravity} =$ mgh, measured from a chosen reference height.

  30. A motorcycle helmet has impact-absorbing foam under a rigid shell. Which design change best reduces impact-force injuries to the rider's head?

    • Increase the foam thickness to increase the time over which the impact force acts
    • Decrease the foam thickness to decrease the time over which the impact force acts
    • Decrease the foam thickness to increase the head's change in speed
    • Increase the foam thickness to decrease the head's change in speed

    Impulse: FΔt = mΔv. The crash fixes the head's change in momentum, so stretching the stopping time Δt lowers the average force F. Thicker foam takes longer to compress.

  31. From 2000 to 2020 the efficiency of U.S. natural-gas power plants rose from about 30% to about 43%. Which statement is correct?

    • The ratio of energy output to energy input increased
    • The energy output became greater than the energy input
    • The energy input and output increased by the same proportion
    • Less energy was wasted as heat because more fuel was burned

    Efficiency = useful energy output ÷ energy input. A higher efficiency means a larger output-to-input ratio; output can never exceed input.

Unit 7: Electricity (28)

  1. Two point charges of $+2 μC$ and $+3 μC$ are separated by 0.1 m. What is the electric force between them (k=8.99×10⁹ N·m²/C²)?

    • Approximately 5.4 N
    • Approximately 54 N
    • Approximately 0.54 N
    • Approximately 540 N

    Fe = kq1q2/r² = (8.99×10⁹)(2×10⁻⁶)(3×10⁻⁶)/(0.1²) = (8.99×10⁹)(6×10⁻¹²)/0.01 ≈ 5.39 N.

  2. If the distance between two charges is halved, the electric force between them becomes:

    • Half as strong
    • Twice as strong
    • 4 times as strong
    • 1/4 as strong

    Coulomb's law is inverse-square: halving r increases force by a factor of (1/0.5)²=4.

  3. Electric field lines around an isolated positive charge point:

    • Toward the charge
    • Away from the charge
    • In circles around the charge
    • In random directions

    Field lines point away from positive charges (the direction a positive test charge would be pushed) and toward negative charges.

  4. A resistor has V=12 V across it and carries I=3 A. What is its resistance?

    • $4 Ω$
    • $36 Ω$
    • $0.25 Ω$
    • $15 Ω$

    $R = V/I = 12/3 = 4 Ω$.

  5. Three $2 Ω$ resistors are connected in series. What is the total resistance?

    • $2 Ω$
    • $0.67 Ω$
    • $6 Ω$
    • $8 Ω$

    Series resistances add directly: Rtotal = $2+2+2 = 6 Ω$.

  6. Three $6 Ω$ resistors are connected in parallel. What is the total resistance?

    • $18 Ω$
    • $6 Ω$
    • $2 Ω$
    • $0.5 Ω$

    1/Rtotal = 1/6+1/6+1/6 = 3/6 = 1/2, so Rtotal = $2 Ω$.

  7. In a series circuit with two resistors, which quantity is the same through both resistors?

    • Voltage
    • Current
    • Power
    • Resistance

    In a series circuit, there's only one path, so the current is identical through every component.

  8. In a parallel circuit, which quantity is the same across every branch?

    • Current
    • Resistance
    • Voltage
    • Power

    Each parallel branch connects across the same two nodes, so voltage is identical across all branches.

  9. A circuit has a 12 V source and total resistance of $4 Ω$. What is the total current?

    • 3 A
    • 48 A
    • 0.33 A
    • 16 A

    I = V/R = 12/4 = 3 A.

  10. A device draws 2 A at 120 V. What power does it consume?

    • 60 W
    • 240 W
    • 122 W
    • 0.017 W

    P = IV = (2)(120) = 240 W.

  11. If one bulb burns out (opens) in a series string of holiday lights, what happens to the rest of the string?

    • The rest stay lit, since current flows around the burned-out bulb
    • The rest also go out, since the circuit is broken
    • The rest get brighter, since the broken bulb no longer uses power
    • The rest get dimmer, since the voltage is shared among fewer bulbs

    In a series circuit there's only one current path; if it's broken anywhere, no current flows anywhere in the loop.

  12. If one branch fails (opens) in a parallel circuit, what happens to the other branches?

    • They stop working too
    • They continue to operate normally
    • They short-circuit
    • Voltage across them becomes zero

    Parallel branches are independent paths; if one opens, current can still flow through the remaining branches.

  13. Which of the following increases the resistance of a wire?

    • Increasing its cross-sectional area
    • Increasing its length
    • Decreasing its length
    • Decreasing its resistivity

    R = ρL/A — resistance increases directly with length and decreases with cross-sectional area.

  14. What is the SI unit of electric charge?

    • Ampere
    • Volt
    • Coulomb
    • Ohm

    The coulomb (C) is the SI unit of electric charge.

  15. A charge of 4 C flows past a point in a wire in 2 seconds. What is the current?

    • 8 A
    • 2 A
    • 0.5 A
    • 6 A

    I = q/t = 4/2 = 2 A.

  16. Doubling the voltage across a fixed resistor does what to the power dissipated?

    • Doubles it
    • Quadruples it
    • Halves it
    • No change

    P = V²/R, so doubling V quadruples P (for constant R).

  17. Two point charges are separated by a distance r. If r is halved, the electric force between them becomes:

    • half as large
    • twice as large
    • four times as large
    • one-fourth as large

    Coulomb's law is inverse-square: halving r multiplies the force by 4.

  18. The electric field points:

    • away from negative charges and toward positive charges
    • away from positive charges and toward negative charges
    • along the direction a magnetic compass needle would point
    • opposite to the force on a positive test charge

    By convention E points the way a positive test charge would be pushed: out of + and into -.

  19. A 12 V battery drives 3.0 A through a resistor. The resistance is:

    • 4.0 ohms
    • 36 ohms
    • 0.25 ohms
    • 15 ohms

    R = V/I = 12/3.0 = 4.0 ohms.

  20. Power dissipated by a resistor carrying 2.0 A with 6.0 V across it is:

    • 3.0 W
    • 12 W
    • 8.0 W
    • 0.33 W

    P = IV = 2.0 x 6.0 = 12 W.

  21. Two 6 ohm resistors are connected in series. The total resistance is:

    • 3 ohms
    • 12 ohms
    • 6 ohms
    • 0.33 ohms

    Series resistances add: 6 + 6 = 12 ohms.

  22. Two 6 ohm resistors are connected in parallel. The total resistance is:

    • 12 ohms
    • 6 ohms
    • 3 ohms
    • 0.33 ohms

    1/R = 1/6 + 1/6 = 1/3, so R = 3 ohms.

  23. In a series circuit, which quantity is the same through every component?

    • Voltage
    • Current
    • Power
    • Resistance

    A series circuit has one path, so the current is identical everywhere.

  24. Adding another resistor in parallel to a circuit causes the total resistance to:

    • increase
    • decrease
    • stay the same
    • become infinite

    Parallel paths give current more ways to flow, lowering total resistance.

  25. Electric current is defined as:

    • charge per unit time
    • voltage per unit resistance times time
    • energy per unit charge
    • force per unit charge

    I = delta q / delta t, measured in amperes (coulombs per second).

  26. Like electric charges:

    • attract
    • repel
    • exert no force
    • always neutralize

    Like charges repel; opposite charges attract.

  27. An ideal voltmeter is connected:

    • in series and has zero resistance
    • in parallel and has very high resistance
    • in series and has very high resistance
    • in parallel and has zero resistance

    A voltmeter measures potential difference across a component, so it is in parallel with high resistance to avoid drawing current.

  28. Charging a balloon by rubbing it on hair transfers:

    • protons from the hair to the balloon
    • electrons between the two surfaces
    • neutrons to the balloon
    • new charge created by friction

    Rubbing moves electrons from one surface to the other; total charge is conserved.

Unit 8: Magnetism & Electromagnetism (26)

  1. Which statement about magnetic poles is correct?

    • Isolated magnetic monopoles have been observed in labs
    • Like poles attract and opposite poles repel
    • Like poles repel and opposite poles attract
    • Magnets only have a single pole

    As with electric charges, like magnetic poles repel and opposite poles attract; monopoles have never been observed.

  2. Outside a bar magnet, magnetic field lines point:

    • From south to north
    • From north to south
    • In random directions
    • Only inward, toward the center

    Outside the magnet, field lines emerge from the north pole and enter the south pole, forming closed loops through the magnet's interior.

  3. A charged particle moves parallel to a magnetic field. What magnetic force does it experience?

    • Maximum force
    • Zero force
    • Force perpendicular to both v and B, of medium size
    • Force equal to qvB

    F = qvB $\sinθ$; when $θ=0°$ (parallel), $\sinθ=0$, so the force is zero.

  4. A charged particle moves perpendicular to a magnetic field. What is the shape of its resulting path?

    • A straight line
    • A parabola
    • A circle
    • An ellipse

    The magnetic force is always perpendicular to velocity, continuously changing direction without changing speed — producing uniform circular motion.

  5. The magnetic force on a moving charge does what to its kinetic energy?

    • Increases it, since the force acts on the charge as it moves
    • Decreases it, since the magnetic force opposes the charge's motion
    • Alternately raises and lowers it, since the force changes sign every cycle
    • Leaves it unchanged, since the force is always perpendicular to velocity

    Since F is always perpendicular to v, it does no work on the charge, so kinetic energy (and speed) stays constant.

  6. What is required to induce an EMF in a loop of wire via electromagnetic induction?

    • A strong, constant magnetic field through the loop
    • A changing magnetic flux through the loop
    • A stationary magnet near the loop
    • Any magnetic field, changing or not

    Faraday's Law requires a CHANGING magnetic flux — a constant field through a stationary loop induces nothing.

  7. According to Lenz's Law, an induced current flows in a direction that:

    • Increases the change in flux that caused it
    • Opposes the change in flux that caused it
    • Has no relationship to the change in flux
    • Is always clockwise

    Lenz's Law: induced current opposes the change in magnetic flux, consistent with conservation of energy.

  8. A generator converts:

    • Electrical energy to mechanical energy
    • Mechanical energy to electrical energy
    • Chemical energy to electrical energy
    • Electrical energy to chemical energy

    A generator uses mechanical motion (rotating a coil in a magnetic field) to induce an EMF — converting mechanical to electrical energy.

  9. A motor converts:

    • Electrical energy to mechanical energy
    • Mechanical energy to electrical energy
    • Thermal energy to electrical energy
    • Light energy to electrical energy

    A motor uses current in a magnetic field to produce a force (torque), converting electrical energy into mechanical motion.

  10. A wire carrying current I=5 A, of length 0.4 m, sits perpendicular to a magnetic field of 0.2 T. What force does it experience?

    • 0.4 N
    • 4 N
    • 1 N
    • 0.04 N

    $F=BIL \sinθ =$ (0.2)(5)(0.4)(sin90°) = 0.4 N.

  11. Increasing the number of turns in a coil used for electromagnetic induction:

    • Decreases the induced EMF
    • Has no effect on the induced EMF
    • Increases the induced EMF
    • Reverses the polarity only

    More turns means the changing flux links more loops, increasing the total induced EMF.

  12. The direction of the magnetic field around a straight current-carrying wire is found using:

    • Snell's Law
    • The right-hand rule
    • Ohm's Law
    • Kepler's Law

    The right-hand rule: point the thumb along conventional current, and the curled fingers show the direction of the circular magnetic field.

  13. A magnet is pushed into a stationary coil of wire, inducing a current. If the magnet is pushed in faster, the induced EMF:

    • Decreases
    • Stays the same
    • Increases
    • Becomes zero

    Faster motion means a faster rate of change of flux, which increases the induced EMF (Faraday's Law).

  14. Why do AC generators produce alternating (rather than direct) current?

    • Because the electrons inside the wire randomly reverse their motion as the coil heats up
    • Because the permanent magnet's field reverses its direction every time the coil turns
    • Because the coil's orientation relative to the magnetic field continuously changes as it rotates
    • Because the wire's resistance alternates between high and low values as the coil spins

    As the coil rotates, the rate and direction of flux change through it varies sinusoidally, producing an alternating induced EMF.

  15. Which of the following experiences zero magnetic force in a magnetic field?

    • A charge moving perpendicular to the field
    • A stationary charge
    • A charge moving at an angle to the field
    • A current-carrying wire perpendicular to the field

    Magnetic force depends on velocity $(F=qvB \sinθ)$; a stationary charge (v=0) experiences no magnetic force.

  16. The magnetic force on a charged particle moving parallel to a magnetic field is:

    • maximum
    • zero
    • directed along the field
    • equal to qE

    F = qvB sin(theta); when v is parallel to B, sin(0) = 0.

  17. A steady magnetic field passes through a stationary wire loop. The induced EMF is:

    • large and constant
    • zero
    • proportional to the field strength
    • alternating

    Induction requires a CHANGING flux; a steady field through a still loop induces nothing.

  18. Lenz's law states that an induced current flows in the direction that:

    • reinforces the change in flux that produced it
    • opposes the change in flux that produced it
    • is always clockwise as seen from the magnet
    • increases the resistance of the circuit

    The induced current opposes the flux change, consistent with energy conservation.

  19. A step-up transformer has more turns on the secondary than the primary. It:

    • raises voltage and lowers current
    • raises both voltage and current
    • lowers voltage and raises current
    • has no effect on voltage

    $V_s/V_p = N_s/N_p$; ideally power is conserved, so higher voltage means lower current.

  20. Magnetic field lines around a straight current-carrying wire form:

    • straight lines along the wire
    • circles around the wire
    • field lines pointing radially outward
    • no pattern

    The right-hand rule gives circular field lines encircling the wire.

  21. The magnetic force on a moving charge does no work because it:

    • is always zero
    • acts perpendicular to the velocity
    • is a conservative force
    • cancels gravity

    A force perpendicular to velocity changes direction but not speed, so it does zero work.

  22. A generator converts:

    • electrical energy to mechanical energy
    • mechanical energy to electrical energy
    • heat to light
    • chemical energy to electrical energy

    Rotating a coil in a magnetic field changes flux and induces an EMF.

  23. Two parallel wires carry current in the same direction. They:

    • attract each other
    • repel each other
    • exert no magnetic force
    • spin

    Parallel currents in the same direction attract; opposite directions repel.

  24. Increasing the number of turns per unit length in a solenoid (same current):

    • weakens the internal field
    • strengthens the internal field
    • reverses the field
    • has no effect

    Solenoid field strength grows with current and with turns per unit length.

  25. A magnetic flux through a loop can change by:

    • changing B alone, since area and angle do not matter
    • rotating the loop alone, since B and area do not matter
    • changing the loop's resistance or the wire's thickness
    • changing B, the loop area, or the loop's orientation

    Flux = B A cos(theta); any of B, A, or theta changing alters the flux.

  26. An electromagnet differs from a permanent magnet mainly because it:

    • is always stronger
    • can be switched on and off
    • has only one pole
    • needs no current

    An electromagnet's field exists only while current flows, so it can be turned off.

Unit 9: Simple Harmonic Motion (12)

  1. For an object in simple harmonic motion, the restoring force is:

    • constant in magnitude and directed toward equilibrium
    • proportional to velocity and directed along the motion
    • proportional to displacement and directed toward equilibrium
    • proportional to displacement and directed away from equilibrium

    SHM is defined by F = -k x, a restoring force growing with displacement.

  2. The period of a mass on a spring is T = 2 pi $\sqrt{m/k}$. Quadrupling the mass:

    • doubles the period
    • halves the period
    • quadruples the period
    • leaves the period unchanged

    T is proportional to sqrt(m), so 4x mass gives 2x period.

  3. The period of a simple pendulum depends on:

    • its mass and length
    • its length and g
    • its amplitude and mass
    • only the mass

    T = 2 pi $\sqrt{L/g}$; mass and (small) amplitude do not matter.

  4. A pendulum clock is taken to the Moon, where g is smaller. The clock will:

    • run fast
    • run slow
    • keep perfect time
    • stop

    Smaller g means a longer period, so each swing takes longer and the clock runs slow.

  5. In SHM, the speed of the oscillator is greatest:

    • at the turning points
    • at the equilibrium position
    • halfway between center and end
    • nowhere; it is constant

    All the energy is kinetic at equilibrium, so the speed peaks there.

  6. In SHM, the acceleration has its largest magnitude:

    • at equilibrium
    • at the maximum displacement
    • when speed is greatest
    • at every point equally

    Acceleration is proportional to displacement, so it is largest at the extremes.

  7. Increasing the amplitude of an ideal mass-spring oscillator changes its period:

    • it increases
    • it decreases
    • it does not change
    • it becomes zero

    The period of ideal SHM is independent of amplitude.

  8. The total mechanical energy of a mass-spring system in SHM is:

    • $0.5 k A^2$
    • 0.5 m A
    • k A
    • m g A

    At the turning point all energy is elastic PE = $0.5 k A^2$, and it stays constant.

  9. Damping in a real oscillator mainly causes the:

    • period to grow quickly
    • amplitude to decrease over time
    • frequency to double
    • restoring force to vanish

    Energy loss to friction/air resistance shrinks the amplitude while barely changing the period.

  10. Resonance occurs when a system is driven at:

    • any frequency
    • one of its natural frequencies
    • a frequency far from its natural frequency
    • zero frequency

    Driving at a natural frequency transfers energy efficiently and builds a large amplitude.

  11. A 0.50 kg mass on a spring with k = 200 N/m has a period of about:

    • 0.31 s
    • 3.1 s
    • 0.10 s
    • 20 s

    T = 2 pi $\sqrt{0.50/200} = 2$ pi $\sqrt{0.0025} = 2$ pi (0.05) = 0.31 s.

  12. Quadrupling the length of a simple pendulum changes its period by a factor of:

    • 4
    • 2
    • 1/2
    • sqrt(2)

    T is proportional to sqrt(L), so 4x length gives 2x period.

Unit 10: Waves & Sound (27)

  1. A wave has a frequency of 5 Hz and a wavelength of 2 m. What is its speed?

    • 2.5 m/s
    • 10 m/s
    • 7 m/s
    • 0.4 m/s

    $v = fλ = (5)(2) = 10 m/s$.

  2. If the frequency of a wave increases while it stays in the same medium, its wavelength:

    • Increases
    • Decreases
    • Stays the same
    • Cannot be determined

    Since $v=fλ$ and speed is fixed by the medium, increasing f must decrease $λ$ proportionally.

  3. Sound is best classified as a:

    • Transverse wave
    • Longitudinal wave
    • Electromagnetic wave
    • Standing wave only

    Sound is a longitudinal wave — particles vibrate parallel to the direction of wave travel, creating compressions and rarefactions.

  4. Sound cannot travel through a vacuum because:

    • Sound waves lose all their energy once there is no air resistance
    • A vacuum is too cold for sound waves to keep vibrating
    • Sound travels as a transverse wave that needs light to carry it
    • Sound requires a medium (particles) to compress and rarefy

    Sound is a mechanical wave requiring particles to transmit compressions and rarefactions; a vacuum has no particles.

  5. Sound travels fastest through which medium?

    • Air
    • Water
    • Steel (a solid)
    • Vacuum

    Sound travels fastest in solids because particles are closest together and transmit vibrations most efficiently, slower in liquids, slowest in gases.

  6. Two identical waves arrive perfectly in phase at the same point. This is an example of:

    • Destructive interference
    • Constructive interference
    • The Doppler effect
    • Refraction

    In-phase waves add their displacements together, producing a larger amplitude — constructive interference.

  7. An ambulance siren sounds higher-pitched as it approaches you and lower-pitched as it drives away. This is:

    • Resonance
    • The Doppler effect
    • Constructive interference
    • Refraction

    The Doppler effect: the source moving toward you compresses wavefronts (higher perceived frequency); moving away stretches them (lower perceived frequency).

  8. What actually happens to the siren's emitted frequency as the ambulance moves toward and then away from a stationary observer?

    • It increases as the ambulance approaches, then decreases as it moves away
    • It decreases as the ambulance approaches, then increases as it moves away
    • It stays exactly the same the whole time — only the perceived frequency changes
    • It rises steadily the whole time as the ambulance's speed increases

    The source's actual emitted frequency never changes; only the frequency perceived by the observer shifts due to relative motion.

  9. Which best describes amplitude?

    • The number of complete cycles passing a point each second
    • The distance between two consecutive crests of the wave
    • The maximum displacement from equilibrium, related to a wave's energy
    • The rate at which the wave travels through its medium

    Amplitude is the maximum displacement from equilibrium and relates to the energy/intensity (loudness for sound) carried by the wave.

  10. A standing wave is formed by:

    • Two identical waves traveling in opposite directions in a confined medium
    • A single wave traveling in one direction through an open medium
    • Two waves of different speeds moving the same way through a medium
    • A wave refracting as it crosses the boundary between two different media

    Standing waves result from the superposition of two identical waves traveling in opposite directions, creating fixed nodes and antinodes.

  11. A wave's period is 0.25 s. What is its frequency?

    • 0.25 Hz
    • 4 Hz
    • 2.5 Hz
    • 40 Hz

    f = 1/T = 1/0.25 = 4 Hz.

  12. Which property of a sound wave corresponds to its perceived pitch?

    • Amplitude
    • Wavelength only, independent of frequency
    • Frequency
    • Speed

    Pitch corresponds directly to frequency — higher frequency sounds are perceived as higher pitch.

  13. Points of maximum displacement in a standing wave are called:

    • Nodes
    • Antinodes
    • Crests only
    • Rarefactions only

    Antinodes are points of maximum displacement; nodes are points of zero displacement.

  14. Resonance occurs when:

    • A wave interferes destructively with itself, producing a large amplitude decrease
    • A system is driven far from its natural frequency, producing a large amplitude increase
    • A system is driven at its natural frequency, producing a large amplitude increase
    • Two waves pass through each other unaffected, producing no change in amplitude

    Resonance is a dramatic amplitude increase that occurs when a driving frequency matches a system's natural frequency.

  15. A wave's amplitude is doubled. Its wave speed:

    • Stays the same, since speed depends on the medium
    • Doubles, since a larger wave carries more energy forward
    • Quadruples, since energy depends on the amplitude squared
    • Is halved, since a bigger wave must travel more slowly

    Wave speed depends only on the properties of the medium, not amplitude or frequency.

  16. A wave has a frequency of 50 Hz and a wavelength of 4.0 m. Its speed is:

    • 12.5 m/s
    • 200 m/s
    • 54 m/s
    • 46 m/s

    v = f lambda = 50 x 4.0 = 200 m/s.

  17. In a transverse wave, the particles of the medium move:

    • parallel to the wave's direction of travel
    • perpendicular to the wave's direction of travel
    • in circles around the source
    • in the same direction as the energy, then back

    Transverse means particle displacement is perpendicular to wave propagation.

  18. Sound is an example of which type of wave?

    • Transverse
    • Longitudinal
    • Electromagnetic
    • Standing only

    Sound is a longitudinal pressure wave; particle motion is parallel to propagation.

  19. When a wave passes from one medium into another, the quantity that stays the same is:

    • speed
    • wavelength
    • frequency
    • amplitude

    Frequency is set by the source and does not change on refraction; speed and wavelength do.

  20. A police siren sounds higher in pitch as the car approaches you. This is:

    • the Doppler effect
    • diffraction
    • total internal reflection
    • resonance

    Approaching motion compresses the wavefronts, raising the observed frequency.

  21. Two sound waves of 256 Hz and 260 Hz are played together. The beat frequency is:

    • 4 Hz
    • 516 Hz
    • 258 Hz
    • 2 Hz

    Beat frequency is the difference: |260 - 256| = 4 Hz.

  22. At a node of a standing wave, the medium:

    • oscillates with maximum amplitude
    • does not move
    • moves at constant speed
    • has maximum kinetic energy

    A node is a point of zero displacement; antinodes have maximum amplitude.

  23. The speed of a wave on a string depends mainly on:

    • the frequency at which the string is shaken
    • the properties of the string (tension and mass per length)
    • the amplitude of the pulse sent along the string
    • the wavelength chosen by the person shaking it

    Wave speed is a property of the medium; changing frequency changes wavelength, not speed.

  24. Increasing the frequency of a wave in a fixed medium causes its wavelength to:

    • increase
    • decrease
    • stay the same
    • become zero

    Since v = f lambda and v is fixed, higher f means shorter lambda.

  25. Sound cannot travel through:

    • water
    • steel
    • air
    • a vacuum

    A mechanical wave like sound needs a medium; a vacuum has none.

  26. A string fixed at both ends vibrates in its fundamental mode. The string length equals:

    • one wavelength
    • half a wavelength
    • two wavelengths
    • a quarter wavelength

    The fundamental fits one half-wavelength between the two fixed ends (nodes).

  27. Loudness of a sound is most directly related to a wave's:

    • frequency
    • amplitude
    • speed
    • wavelength

    Greater amplitude carries more energy, heard as greater loudness; frequency sets pitch.

Unit 11: Optics (27)

  1. A light ray strikes a plane mirror at 30° from the normal. At what angle does it reflect?

    • 30° from the normal
    • 60° from the normal
    • 30° from the mirror surface
    • 0°

    Law of reflection: angle of incidence = angle of reflection, both measured from the normal, so it reflects at 30° from the normal.

  2. A plane mirror image is:

    • Real, inverted, same size as the object
    • Virtual, inverted, same size as the object
    • Virtual, upright, same size as the object
    • Real, upright, magnified compared to the object

    A flat (plane) mirror always forms a virtual, upright, same-size image located as far behind the mirror as the object is in front.

  3. Light travels from air (n=1.00) into glass (n=1.50). It will bend:

    • Away from the normal
    • Toward the normal
    • Not at all
    • Parallel to the surface

    Light bends toward the normal when entering a medium with a higher index of refraction (denser, slower light speed).

  4. Light in water (n=1.33) hits the water-air boundary at an angle greater than the critical angle. What happens?

    • Some light refracts into the air, bending toward the normal as it exits
    • The light is totally absorbed by the surface and converted into heat
    • The light disperses into its colors and spreads out into the air
    • Total internal reflection occurs — all light reflects back into the water

    Beyond the critical angle, going from a denser to less dense medium, total internal reflection occurs — no light exits.

  5. Using Snell's Law, light passes from a medium with n1=1.5 into a medium with n2=1.0, hitting the boundary at 30° from the normal. Which best describes the refracted angle?

    • Less than 30°, bending toward the normal at the boundary
    • Exactly 30°, passing straight through with no bending
    • Greater than 30°, bending away from the normal
    • The light cannot refract, since 30° exceeds the critical angle

    $n1sinθ1=n2sinθ2$ → since n2<n1, $\sinθ2$ must be larger than $\sinθ1$, so $θ2>30°$ — light bends away from the normal entering a less dense medium.

  6. A convex (converging) lens forms what kind of image when the object is placed beyond the focal point?

    • Virtual and upright
    • Real and inverted
    • Virtual and inverted
    • No image forms

    When do>f for a converging lens, light rays converge on the far side, forming a real, inverted image.

  7. An object is placed inside the focal length of a convex lens (do<f). The resulting image is:

    • Real, inverted, reduced
    • Virtual, upright, magnified
    • Real, upright, magnified
    • Virtual, inverted, reduced

    When the object is closer than the focal point, the lens acts as a magnifying glass, producing a virtual, upright, magnified image.

  8. An object is placed 30 cm from a convex lens with a focal length of 10 cm. Using 1/f=1/do+1/di, what is the image distance?

    • 7.5 cm
    • 15 cm
    • 20 cm
    • 30 cm

    1/di = 1/f − 1/do = 1/10 − 1/30 = 3/30 − 1/30 = 2/30 = 1/15, so di = 15 cm.

  9. A convex mirror (like a car's passenger side mirror) always forms an image that is:

    • Real, inverted, magnified
    • Virtual, upright, reduced
    • Real, upright, same size
    • Virtual, inverted, magnified

    Convex (diverging) mirrors always produce virtual, upright, reduced images, giving a wider field of view.

  10. Which type of lens or mirror can form a REAL image?

    • Convex (diverging) mirror, if the object is beyond the focal point
    • Concave (converging) mirror, if the object is inside the focal point
    • Concave (converging) mirror, if the object is beyond the focal point
    • Convex (diverging) lens, if the object is placed close to the lens

    Only converging elements (convex lens, concave mirror) can produce real images, and only when the object is beyond the focal point.

  11. What is the index of refraction of a medium in which light travels at 2.0×10⁸ m/s (c=3.0×10⁸ m/s)?

    • 0.67
    • 1.5
    • 1.0
    • 2.0

    n = c/v = (3.0×10⁸)/(2.0×10⁸) = 1.5.

  12. Which best explains why a straw appears bent at the water's surface when viewed from the side?

    • Reflection — light bounces off the water's surface back to your eye
    • Diffraction — light spreads out as it passes around the edge of the straw
    • Refraction — light bends as it passes from water into air
    • Total internal reflection — light is trapped inside the water

    Refraction bends light rays as they cross the water-air boundary due to the change in speed, making the submerged part of the straw appear displaced/bent.

  13. A magnification of m=−2 for an image means:

    • The image is upright and twice the size
    • The image is inverted and twice the size
    • The image is upright and half the size
    • The image is inverted and half the size

    A negative m indicates an inverted image; the magnitude (2) means the image is twice the height of the object.

  14. Diffuse reflection occurs because:

    • The surface is very smooth, keeping parallel rays parallel after bouncing
    • The light is absorbed and re-emitted at a longer wavelength
    • The surface is rough, scattering parallel rays in many directions
    • The law of reflection fails on rough surfaces, so rays leave at random angles

    A rough surface reflects parallel incoming rays at many different angles (still following the law of reflection at each microscopic point), preventing a clear image from forming.

  15. The critical angle for total internal reflection depends on:

    • The ratio of the two media's indices of refraction
    • The brightness of the incoming beam striking the boundary
    • The thickness of the denser medium the light travels through
    • The speed of the observer relative to the boundary

    $\sinθc = n2/n1$, so the critical angle depends on the ratio of the refractive indices of the two media.

  16. According to the law of reflection, the angle of incidence is measured from the:

    • mirror surface
    • normal to the surface
    • reflected ray
    • horizontal

    Both incidence and reflection angles are measured from the normal, and they are equal.

  17. A plane mirror forms an image that is:

    • real, inverted, and the same size
    • virtual, upright, and the same size
    • virtual, inverted, and the same size
    • real, upright, and magnified in size

    A flat mirror produces a virtual, upright, same-size image as far behind as the object is in front.

  18. Light slows down when it enters a medium with a higher index of refraction and bends:

    • away from the normal
    • toward the normal
    • back on itself
    • not at all

    Going into a slower (denser) medium, the ray bends toward the normal.

  19. The index of refraction of a medium is defined as:

    • $v_{medium} / c$
    • $c / v_{medium}$
    • $c x v_{medium}$
    • f x lambda

    $n = c/v_{medium}$, so a larger n means slower light and n is at least 1.

  20. A converging (convex) lens forms a real, inverted image when the object is:

    • at the focal point
    • inside the focal length
    • beyond the focal point
    • touching the lens

    Objects beyond the focal length of a converging lens produce real, inverted images.

  21. A diverging (concave) lens always forms an image that is:

    • real and inverted
    • virtual, upright, and reduced
    • real and magnified
    • the same size as the object

    A diverging lens can only make virtual, upright, smaller images.

  22. Total internal reflection can occur only when light travels from a medium of:

    • high index to low index, beyond the critical angle
    • low index to high index, beyond the critical angle
    • equal indices, at any angle of incidence
    • vacuum into glass, at a steep angle of incidence

    TIR requires going toward a less dense medium at an angle greater than the critical angle.

  23. When white light passes through a prism and spreads into colors, this is called:

    • diffraction
    • dispersion
    • polarization
    • reflection

    Dispersion results because the index of refraction varies slightly with wavelength.

  24. All electromagnetic waves in a vacuum travel at:

    • 343 m/s
    • $3.0 x 10^8 m/s$
    • depends on frequency
    • $1.5 x 10^8 m/s$

    Every EM wave travels $at c = 3.0 x 10^8 m/s$ in vacuum, regardless of frequency.

  25. Which list orders electromagnetic waves from longest to shortest wavelength?

    • gamma, X-ray, visible, radio
    • radio, infrared, visible, ultraviolet
    • ultraviolet, visible, infrared, radio
    • visible, radio, X-ray, gamma

    Radio has the longest wavelength; then infrared, visible, ultraviolet, X-ray, gamma.

  26. When light refracts entering water, which property does NOT change?

    • speed
    • wavelength
    • direction
    • frequency

    Frequency (and color) is fixed by the source; speed, wavelength, and direction change.

  27. A double-slit experiment produces alternating bright and dark bands because of:

    • reflection
    • interference of light waves
    • total internal reflection
    • the photoelectric effect

    Light waves from the two slits interfere constructively (bright) and destructively (dark).

Unit 12: Modern Physics (27)

  1. In the photoelectric effect, increasing the intensity of light above the threshold frequency mainly increases:

    • The maximum kinetic energy of ejected electrons
    • The number of ejected electrons
    • The threshold frequency
    • The work function of the metal

    Intensity relates to the number of photons hitting the metal per second, which increases the number (rate) of ejected electrons, not their individual energy.

  2. In the photoelectric effect, increasing the frequency of light (above threshold) mainly increases:

    • The maximum kinetic energy of ejected electrons
    • The number of electrons ejected each second
    • The work function of the metal being illuminated
    • The threshold frequency of the metal surface

    Each photon's energy is E=hf; higher frequency means each photon carries more energy, increasing the max KE of ejected electrons (KEmax=hf−W0).

  3. A metal has a work function of 2.0 eV. Light with photon energy 3.0 eV strikes it. What is the maximum kinetic energy of an ejected electron?

    • 5.0 eV
    • 1.0 eV
    • 2.0 eV
    • 3.0 eV

    KEmax = hf − W0 = 3.0 − 2.0 = 1.0 eV.

  4. Light striking a metal below its threshold frequency, no matter how intense, will:

    • Eject electrons with high kinetic energy
    • Eject many electrons but with low kinetic energy
    • Eject no electrons at all
    • Eject electrons only after a time delay

    Below the threshold frequency, individual photons don't carry enough energy to overcome the work function, so no electrons are ejected regardless of intensity.

  5. The de Broglie wavelength of a particle is given by:

    • $λ =$ hf
    • $λ = h/p$
    • $λ =$ hc
    • $λ = pc/h$

    De Broglie's relation: $λ = h/p = h/(mv)$, describing the wave nature of matter.

  6. Which experiment provides strong evidence for the wave nature of light?

    • Photoelectric effect threshold frequency
    • Compton scattering of X-rays by electrons
    • Discrete emission lines from hot gases
    • Double-slit diffraction/interference

    Double-slit diffraction and interference patterns are classic evidence of light's wave nature; the photoelectric effect instead evidences light's particle nature.

  7. In the Bohr model, an atom emits a photon when:

    • An electron jumps from a lower to a higher energy level
    • An electron falls from a higher to a lower energy level
    • An electron is captured by the nucleus of the atom
    • An electron stays in one stationary orbit for a long time

    A photon is emitted when an electron transitions from a higher to a lower allowed energy level, releasing energy equal to the level difference.

  8. Bright-line emission spectra are unique to each element because:

    • Each element's electrons orbit at continuous, unrestricted energy values
    • Each element has electrons that absorb and emit every possible wavelength
    • Each element's spectrum is set by how hot the gas sample happens to be
    • Each element has its own unique set of quantized electron energy levels

    Each element has a distinct set of quantized energy levels, so the photon energies (and thus wavelengths) it can emit/absorb form a unique 'fingerprint.'

  9. Which particle is emitted in alpha decay?

    • A hydrogen nucleus (1 proton + 0 neutrons)
    • A high-energy electron from the nucleus
    • A single neutron ejected from the nucleus
    • A helium nucleus (2 protons + 2 neutrons)

    Alpha decay emits an alpha particle, which is a helium-4 nucleus (2 protons, 2 neutrons).

  10. After an alpha decay, the mass number and atomic number of the parent nucleus change by:

    • Mass number −4, atomic number −2
    • Mass number −2, atomic number −4
    • Mass number unchanged, atomic number −2
    • Mass number −4, atomic number unchanged

    Alpha decay removes 2 protons and 2 neutrons, so mass number decreases by 4 and atomic number decreases by 2.

  11. Which type of radioactive decay is stopped by just a sheet of paper?

    • Alpha
    • Beta
    • Gamma
    • All types equally

    Alpha particles have the least penetrating power (but highest ionizing power) and are stopped by paper or skin.

  12. Which type of radioactive decay requires lead or thick concrete to stop?

    • Alpha
    • Beta
    • Gamma
    • None; all pass through lead

    Gamma rays are high-energy photons with the greatest penetrating power, requiring dense shielding like lead or thick concrete.

  13. A radioactive sample has a half-life of 4 days. After 12 days, what fraction of the original sample remains?

    • 1/2
    • 1/4
    • 1/8
    • 1/16

    12 days = 3 half-lives, so the remaining fraction is (1/2)³ = 1/8.

  14. Nuclear fusion, the process that powers the Sun, involves:

    • Splitting a heavy nucleus into lighter nuclei, releasing energy
    • Combining light nuclei into a heavier nucleus, releasing energy
    • Combining heavy nuclei into a lighter nucleus, absorbing energy
    • Converting electrons into protons inside a hot, dense plasma

    Fusion combines light nuclei (e.g., hydrogen into helium), releasing a large amount of energy — the process that powers stars.

  15. According to mass-energy equivalence, a small amount of mass converted entirely to energy releases:

    • A tiny amount of energy, since the mass involved is so small
    • Energy that appears as heat and cannot do any other work
    • No energy at all, since mass and energy are separate quantities
    • An amount of energy given by E=mc², which is very large due to c²

    E=mc² shows that even a tiny mass corresponds to an enormous amount of energy because c² (≈9×10¹⁶ m²/s²) is such a large number.

  16. Beta decay involves:

    • A neutron converting to a proton (or vice versa), emitting an electron or positron
    • Emission of a helium nucleus, lowering the atomic number by 2 and mass number by 4
    • Emission of a high-energy photon, with the nucleus dropping to a lower energy state
    • Splitting of a heavy nucleus into two medium nuclei, releasing neutrons and energy

    In beta-minus decay, a neutron converts to a proton and an electron (beta particle) is emitted; atomic number increases by 1, mass number stays the same.

  17. The energy of a photon is given by:

    • $E = mc^2$
    • E = hf
    • $E = 0.5 m v^2$
    • E = qV

    Photon energy E = hf = hc/lambda, with h Planck's constant.

  18. In the photoelectric effect, electrons are ejected from a metal only if the light's:

    • intensity exceeds a threshold value
    • wavelength exceeds a threshold value
    • polarization matches the metal's surface
    • frequency exceeds a threshold value

    Below the threshold frequency, no electrons are ejected no matter how bright the light.

  19. Increasing the frequency of light above the photoelectric threshold increases the:

    • maximum kinetic energy of ejected electrons
    • number of electrons ejected per second
    • work function of the metal surface
    • speed of the light in a vacuum

    $KE_{max} =$ hf - W, so higher f gives each electron more kinetic energy.

  20. An atom emits light of a specific wavelength when an electron:

    • drops from a higher energy level to a lower one
    • jumps from a lower energy level to a higher one
    • is added to the nucleus of the atom
    • is removed from the atom's outermost shell

    The photon energy equals the difference between the two electron energy levels.

  21. In alpha decay, the atomic number of the nucleus:

    • increases by 1
    • decreases by 2
    • stays the same
    • increases by 2

    An alpha particle is a helium-4 nucleus, so the parent loses 2 protons (and 2 neutrons).

  22. The half-life of a radioactive isotope is the time for:

    • all of the sample to decay
    • half of the sample to decay
    • the sample to double
    • the decay rate to double

    After one half-life, half the original nuclei remain; it is constant for a given isotope.

  23. $E = mc^2$ tells us that:

    • mass and energy are two unrelated quantities
    • energy is measured without any units at all
    • the speed of light depends on an object's mass
    • a small mass corresponds to a large energy

    Because $c^2$ is very large, converting a tiny amount of mass releases enormous energy.

  24. Nuclear fusion involves:

    • splitting a heavy nucleus into lighter ones
    • combining light nuclei into a heavier one
    • an electron falling to a lower level
    • a proton turning into a neutron

    Fusion joins light nuclei (like hydrogen isotopes) and powers the Sun.

  25. A beta-minus decay converts a neutron into a:

    • proton, emitting an electron
    • proton, emitting a photon
    • helium nucleus
    • positron only

    In beta-minus decay a neutron becomes a proton and an electron is emitted, raising the atomic number by 1.

  26. Each chemical element produces a unique:

    • half-life
    • line spectrum
    • mass number
    • index of refraction

    The spacing of an element's energy levels gives it a characteristic set of spectral lines.

  27. Gamma decay changes a nucleus's:

    • atomic number by 1 but not its mass number
    • energy, but not its atomic or mass number
    • mass number by 4 and atomic number by 2
    • charge by 2 and its number of neutrons by 2

    A gamma ray is a photon; the nucleus loses energy but keeps the same protons and neutrons.

Hard Mode Questions — 171 questions
Unit 1: Measurement, Vectors & Motion Basics (25)

  1. A boat that can travel 10 km/h in still water is aimed straight across a river with a 4.0 km/h current. Its speed relative to the shore is closest to:

    • 14 km/h
    • 6.0 km/h
    • 10.8 km/h
    • 9.2 km/h

    √(10² + 4.0²) = √116 ≈ 10.8 km/h.

  2. The same boat crosses a river 3.0 km wide. How far downstream does the current carry it by the time it reaches the far bank?

    • 3.0 km
    • 1.2 km
    • 4.0 km
    • 0.75 km

    Crossing time = 3.0 km / 10 km/h = 0.30 h; downstream = 4.0 km/h × 0.30 h = 1.2 km.

  3. Forces of 20. N east and 15 N north act on a point. The resultant is:

    • 25 N at about 37° north of east
    • 35 N at 37° north of east
    • 25 N at 53° north of east
    • 5.0 N at 37° north of east

    R = √(20² + 15²) = 25 N and θ = tan⁻¹(15/20) ≈ 37° from the east direction.

  4. A 50 N force acts 37° above the horizontal. Its horizontal and vertical components are approximately:

    • 30 N and 40 N
    • 50 N and 37 N
    • 25 N and 25 N
    • 40 N and 30 N

    50 cos 37° ≈ 40 N and 50 sin 37° ≈ 30 N.

  5. A hiker walks 6 km east, 8 km north, 6 km west, and 8 km south. Her total distance and displacement are:

    • 0 km and 28 km
    • 28 km and 28 km
    • 28 km and 0 km
    • 14 km and 0 km

    She walks 6 + 8 + 6 + 8 = 28 km but returns to her start.

  6. A runner goes 3.0 km north in 15 minutes and then 4.0 km east in 20 minutes. Her average speed for the trip is:

    • 8.6 km/h
    • 12 km/h
    • 7.0 km/h
    • 20 km/h

    Distance 7.0 km in 35 min = 7.0 / (35/60) = 12 km/h.

  7. For the same run (3.0 km north, then 4.0 km east, 35 minutes total), the magnitude of her average velocity is:

    • 8.6 km/h
    • 12 km/h
    • 5.0 km/h
    • 14 km/h

    Displacement = √(3² + 4²) = 5.0 km in 35/60 h, so 5.0 / 0.583 ≈ 8.6 km/h.

  8. Light travels at 3.0 × 10^8 m/s. How far does it go in 2.0 × 10^3 s?

    • 6.0 × 10^5 m
    • 1.5 × 10^5 m
    • 6.0 × 10^24 m
    • 6.0 × 10^11 m

    (3.0 × 10^8)(2.0 × 10^3) = 6.0 × 10^11 m.

  9. Sunlight travels 1.5 × 10^11 m to reach Earth at 3.0 × 10^8 m/s. How long does the trip take?

    • 5.0 × 10^-3 s
    • 4.5 × 10^19 s
    • 5.0 × 10^2 s
    • 5.0 × 10^19 s

    t = d/v = (1.5 × 10^11)/(3.0 × 10^8) = 0.50 × 10^3 = 5.0 × 10^2 s.

  10. Earth's radius is 6.4 × 10^6 m. Its order of magnitude is:

    • 10^6 m
    • 10^7 m
    • 10^5 m
    • 10^8 m

    6.4 is closer to 10 than to 1, so round up to 10^7 m.

  11. An X-ray wavelength of 350 pm is equal to:

    • 3.5 × 10^-10 m
    • 3.5 × 10^-12 m
    • 3.5 × 10^-9 m
    • 3.5 × 10^-7 m

    pico = 10^-12, so 350 pm = 350 × 10^-12 m = 3.5 × 10^-10 m.

  12. A student measures g as 9.60 m/s² when the accepted value is 9.81 m/s². The percent error is about:

    • 0.21%
    • 21%
    • 2.2%
    • 2.1%

    |9.60 − 9.81| / 9.81 × 100% = 0.21 / 9.81 × 100% ≈ 2.1%.

  13. Two forces of 8.0 N and 6.0 N act concurrently on an object. Which resultant is NOT possible?

    • 2.0 N
    • 7.0 N
    • 15 N
    • 14 N

    The resultant must lie between 8.0 − 6.0 = 2.0 N and 8.0 + 6.0 = 14 N.

  14. A 200 N block sits on a 30° frictionless incline. The component of its weight perpendicular to the surface is about:

    • 100 N
    • 173 N
    • 200 N
    • 87 N

    200 cos 30° ≈ 173 N (the parallel component is 200 sin 30° = 100 N).

  15. Forces of 30. N east and 40. N north act on an object. The equilibrant is:

    • 50. N pointing southwest
    • 50. N pointing northeast
    • 70. N pointing southwest
    • 10. N pointing south

    The resultant is 50 N to the northeast, and the equilibrant is equal and opposite.

  16. A 10.0 m displacement at 30° above the +x axis is followed by a 5.0 m displacement along +x. The magnitude of the total displacement is about:

    • 15.0 m
    • 13.7 m
    • 5.0 m
    • 14.5 m

    x = 10 cos 30° + 5.0 ≈ 13.66 m, y = 10 sin 30° = 5.0 m, and √(13.66² + 5.0²) ≈ 14.5 m.

  17. A velocity of 3 m/s east is multiplied by the scalar −2. The result is:

    • 6 m/s east
    • 1.5 m/s west
    • 6 m/s west
    • −6 m/s east

    A negative scalar reverses the direction, and the factor 2 doubles the magnitude.

  18. A car covers the first half of a trip at 30 m/s and the second half of the same distance at 60 m/s. Its average speed is:

    • 45 m/s
    • 40 m/s
    • 30 m/s
    • 60 m/s

    For distance d each way, t = d/30 + d/60 = d/20, so the average is 2d / (d/20) = 40 m/s, not the simple mean.

  19. A runner completes 2.5 laps of a 400 m circular track, starting at the start line (she ends on the opposite side of the track). Her distance and the magnitude of her displacement are about:

    • 1000 m and 127 m
    • 1000 m and 0 m
    • 1000 m and 400 m
    • 127 m and 1000 m

    Distance = 2.5 × 400 = 1000 m. After a half lap she is at the opposite end of a diameter: d = C/π = 400/π ≈ 127 m.

  20. Two vectors of magnitudes 4 and 5 make a 60° angle. Their dot product is:

    • 20
    • 17
    • 0
    • 10

    AB cosθ = 4 × 5 × cos 60° = 20 × 0.5 = 10.

  21. A car moving at 25 m/s brakes with a constant deceleration of 5.0 m/s². Its stopping distance is closest to:

    • 63 m
    • 125 m
    • 5.0 m
    • 31 m

    v_f² = v_0² + 2ad with v_f = 0: d = v_0²/(2a) = 625 ÷ 10. = 62.5 ≈ 63 m.

  22. A v–t graph shows a car at a steady 10. m/s for 4.0 s, then slowing uniformly to rest over the next 2.0 s. The total distance traveled is:

    • 50. m
    • 40. m
    • 60. m
    • 20. m

    Area = rectangle (10. × 4.0 = 40. m) + triangle (½ × 2.0 × 10. = 10. m) = 50. m.

  23. On a position–time graph, the curve rises, levels off, falls, levels off again, and then rises. While the curve is falling, the object is:

    • moving in the negative direction
    • at rest
    • moving in the positive direction and slowing
    • accelerating at a constant positive rate

    A falling curve has a negative slope, so the velocity is negative: the object is moving back toward the origin. The level spots are where it stops and turns around.

  24. A 5.0 N force and a 7.0 N force act concurrently on a point. Which value could NOT be the magnitude of their resultant?

    • 13 N
    • 2.0 N
    • 7.0 N
    • 10. N

    The resultant must lie between |7.0 − 5.0| = 2.0 N and 7.0 + 5.0 = 12.0 N. 13 N is outside that range.

  25. A velocity vector of 25 m/s has a horizontal component of 20. m/s. Its vertical component is:

    • 15 m/s
    • 5.0 m/s
    • 45 m/s
    • 32 m/s

    The components are the legs of a right triangle: √(25² − 20.²) = √(625 − 400) = √225 = 15 m/s.

Unit 2: Kinematics (18)

  1. A car accelerates uniformly from 8.0 m/s to 26 m/s while covering 102 m. What is its acceleration?

    • $3.0 m/s^2$
    • $2.0 m/s^2$
    • $1.5 m/s^2$
    • $5.0 m/s^2$

    $v^2 = v0^2 +$ 2a*dx: $26^2 = 8^2 + 2a(102)$ -> 676 - 64 = 204a -> $a = 3.0 m/s^2$.

  2. A ball is thrown straight up $at 24.5 m/s (g = 9.8 m/s^2)$. What maximum height does it reach?

    • 30.6 m
    • 24.5 m
    • 61.3 m
    • 12.3 m

    At the top v = 0: $0 = 24.5^2 - 2(9.8)h$ -> h = 600.25/19.6 = 30.6 m.

  3. An object starts at x = +5 m moving at -3 m/s with acceleration $+2 m/s^2$. Where is it at t = 4 s?

    • -9 m
    • +9 m
    • +13 m
    • +21 m

    x = 5 + (-3)(4) + 0.5(2)(16) = 5 - 12 + 16 = +9 m.

  4. A stone dropped into a well hits the water 2.0 s later ($g = 9.8 m/s^2$, ignore sound travel). How deep is the well?

    • 9.8 m
    • 19.6 m
    • 39.2 m
    • 4.9 m

    $d = 0.5(9.8)(2.0)^2 = 19.6 m$.

  5. On a velocity-time graph a line goes from (0 s, 4 m/s) to (5 s, 14 m/s). What is the displacement over that interval?

    • 45 m
    • 50 m
    • 70 m
    • 25 m

    Displacement is the trapezoid area: 0.5(4 + 14)(5) = 45 m.

  6. A projectile is launched at 30 m/s at 37 degrees above horizontal $(sin37 = 0.60, cos37 = 0.80, g = 10 m/s^2)$. What is its time of flight on level ground?

    • 1.8 s
    • 3.6 s
    • 2.4 s
    • 6.0 s

    v0y = 30(0.60) = 18 m/s; time up = 1.8 s; total = 3.6 s.

  7. For that projectile (v0 = 30 m/s at 37 deg, $g = 10 m/s^2$), what is the horizontal range on level ground?

    • 86 m
    • 108 m
    • 54 m
    • 72 m

    v0x = 30(0.80) = 24 m/s; range = 24 x 3.6 = 86.4 m.

  8. A car traveling at 20 m/s brakes with constant deceleration and stops in 4.0 s. How far does it travel while stopping?

    • 80 m
    • 40 m
    • 20 m
    • 10 m

    Average velocity = (20 + 0)/2 = 10 m/s; distance = 10 x 4.0 = 40 m.

  9. Two balls are released from the same height: one dropped, one thrown horizontally at 12 m/s. Which lands first?

    • The dropped ball
    • The thrown ball
    • They land at the same time
    • Cannot be determined

    Vertical motion is identical (same height, same g, zero initial vertical speed), so they land together.

  10. An object moving in the +x direction has a negative acceleration. It is:

    • speeding up
    • slowing down
    • moving in the -x direction
    • at rest

    Velocity is positive and acceleration is negative (opposite signs), so the object slows down.

  11. A rocket rises from rest with $a = 20 m/s^2$ for 5.0 s, then the engine cuts off $(g = 10 m/s^2)$. What is its speed just as the engine stops?

    • 50 m/s
    • 100 m/s
    • 150 m/s
    • 200 m/s

    v = a t = 20 x 5.0 = 100 m/s at engine cutoff.

  12. After that engine cutoff at 100 m/s upward $(g = 10 m/s^2)$, how much higher does the rocket coast?

    • 250 m
    • 500 m
    • 1000 m
    • 100 m

    $0 = 100^2 - 2(10)h$ -> h = 10000/20 = 500 m.

  13. A particle's position is $x = 3t^2$ (meters, seconds). Its velocity at t = 2 s is:

    • 6 m/s
    • 12 m/s
    • 24 m/s
    • 3 m/s

    v = dx/dt = 6t; at t = 2 s, v = 12 m/s.

  14. A ball thrown straight down at 5.0 m/s from a 40 m building hits the ground at what speed $(g = 10 m/s^2)?$

    • 25 m/s
    • 28.7 m/s
    • 20 m/s
    • 45 m/s

    $v^2 = 5^2 + 2(10)(40) = 25 + 800 = 825$ -> v = 28.7 m/s.

  15. A car covers the first half of a trip at 30 km/h and the second half (equal distance) at 60 km/h. Its average speed is:

    • 45 km/h
    • 40 km/h
    • 50 km/h
    • 36 km/h

    For equal distances, average speed = 2(30)(60)/(30+60) = 3600/90 = 40 km/h.

  16. A ball is thrown straight up at 20. m/s. Ignoring air resistance (g = 9.8 m/s²), its maximum height is closest to:

    • 20. m
    • 41 m
    • 10. m
    • 2.0 m

    At the top v = 0, so 0 = v_0² − 2gh and h = v_0²/(2g) = 400 ÷ 19.6 ≈ 20. m. 41 m forgets the 2 in 2g.

  17. You hold a bill so your fingers are at its middle; it must fall 7.8 cm before it slips through. With g = 9.8 m/s² and a typical 0.20 s reaction time, can you catch it?

    • No: it falls through in about 0.13 s
    • Yes: it takes about 0.40 s to fall through
    • Yes: it takes about 0.20 s, exactly your reaction time
    • No: it falls through in about 0.016 s

    t = √(2d/g) = √(2 × 0.078 ÷ 9.8) = √0.0159 ≈ 0.13 s, shorter than a 0.20 s reaction time, so the bill is gone before you pinch.

  18. A stone is thrown straight down from a cliff at 5.0 m/s. Ignoring air resistance (g = 9.8 m/s²), its speed after 3.0 s is closest to:

    • 34 m/s
    • 29 m/s
    • 44 m/s
    • 25 m/s

    v_f = v_0 + gt = 5.0 + (9.8)(3.0) = 5.0 + 29.4 ≈ 34 m/s. 29 m/s ignores the starting speed.

Unit 3: Newton's Laws & Dynamics (16)

  1. A 4.0 kg block on a frictionless surface is pulled by a 12 N force at 60 degrees above horizontal. What is its horizontal acceleration (cos60 = 0.5)?

    • $1.5 m/s^2$
    • $3.0 m/s^2$
    • $0.75 m/s^2$
    • $2.6 m/s^2$

    Horizontal force = 12 cos60 = 6.0 N; $a = 6.0/4.0 = 1.5 m/s^2$.

  2. A 10 kg box is pushed across a floor with $mu_k = 0.25$ by a 40 N horizontal force $(g = 10 m/s^2)$. Its acceleration is:

    • $1.5 m/s^2$
    • $4.0 m/s^2$
    • $2.5 m/s^2$
    • $0.5 m/s^2$

    Friction = 0.25(100) = 25 N; net = 40 - 25 = 15 N; $a = 15/10 = 1.5 m/s^2$.

  3. In an elevator accelerating upward $at 2.0 m/s^2, a 60$ kg person stands on a scale $(g = 10 m/s^2)$. The scale reads:

    • 600 N
    • 720 N
    • 480 N
    • 120 N

    N - mg = ma -> N = 60(10 + 2.0) = 720 N.

  4. Two blocks, 3.0 kg and 2.0 kg, are connected by a string on a frictionless table and pulled by 10 N on the 2.0 kg block. The tension in the connecting string is:

    • 4.0 N
    • 6.0 N
    • 10 N
    • 2.0 N

    $a = 10/5.0 = 2.0 m/s^2$; tension pulls the 3.0 kg block: T = 3.0 x 2.0 = 6.0 N.

  5. A 5.0 kg mass hangs from a rope in an elevator that accelerates downward $at 3.0 m/s^2 (g = 10 m/s^2)$. The rope tension is:

    • 50 N
    • 35 N
    • 65 N
    • 15 N

    mg - T = ma -> T = 5.0(10 - 3.0) = 35 N.

  6. A block sits on a 30 degree incline with $mu_s = 0.40$. Will it start to slide (tan30 = 0.58)?

    • Yes, because tan30 > $mu_s$
    • No, because tan30 < $mu_s$
    • Only if pushed
    • Cannot tell without the mass

    It slides when tan(theta) exceeds $mu_s$; 0.58 > 0.40, so it slides.

  7. A 2.0 kg object experiences forces of 8.0 N east and 6.0 N north. Its acceleration magnitude is:

    • $7.0 m/s^2$
    • $5.0 m/s^2$
    • $10 m/s^2$
    • $3.5 m/s^2$

    Net force = $\sqrt{8^2 + 6^2} = 10 N$; $a = 10/2.0 = 5.0 m/s^2$.

  8. A 1200 kg car rounds a flat curve of radius 50 m at 15 m/s. The minimum coefficient of friction needed is $(g = 10 m/s^2)$:

    • 0.30
    • 0.45
    • 0.15
    • 0.60

    mu = $v^2/(g r) = 225/(10 x 50) = 0.45$.

  9. A 6.0 kg block on a frictionless 37 degree incline (sin37 = 0.60) is held by a rope parallel to the incline. The rope tension is $(g = 10 m/s^2)$:

    • 36 N
    • 48 N
    • 60 N
    • 30 N

    T = mg sin(theta) = 6.0(10)(0.60) = 36 N.

  10. A horizontal force accelerates a 3.0 kg block $at 2.0 m/s^2$ across a surface where friction is 4.0 N. The applied force is:

    • 6.0 N
    • 10 N
    • 2.0 N
    • 14 N

    $F_{applied} - 4.0 = 3.0(2.0) = 6.0$ -> $F_{applied} = 10 N$.

  11. An Atwood machine has masses 5.0 kg and 3.0 kg over a frictionless pulley $(g = 10 m/s^2)$. The acceleration of the system is:

    • $2.5 m/s^2$
    • $1.25 m/s^2$
    • $5.0 m/s^2$
    • $10 m/s^2$

    $a = (m1 - m2)g/(m1 + m2) = (2.0)(10)/8.0 = 2.5 m/s^2$.

  12. For that Atwood machine (5.0 kg and 3.0 kg, $a = 2.5 m/s^2, g = 10$), the string tension is:

    • 37.5 N
    • 30 N
    • 50 N
    • 25 N

    For the 3.0 kg mass: T - mg = ma -> T = 3.0(10 + 2.5) = 37.5 N.

  13. A person in an elevator feels heavier than normal. The elevator is:

    • moving upward at a constant speed of any size
    • accelerating downward or slowing while moving up
    • in free fall with the cable broken
    • accelerating upward or slowing while moving down

    An upward acceleration increases the normal force, so the apparent weight rises.

  14. A 20 N crate is on a rough floor. A 6 N horizontal push does not move it. Static friction on the crate is:

    • 6 N
    • 20 N
    • 0 N
    • $mu_s$ times 20 N

    Static friction matches the applied force to keep the crate in equilibrium: 6 N.

  15. A 1000 kg car accelerates from 0 to 20 m/s in 8.0 s on a level road. The net forward force is:

    • 2500 N
    • 20000 N
    • 1250 N
    • 160 N

    $a = 20/8.0 = 2.5 m/s^2$; F = 1000 x 2.5 = 2500 N.

  16. A block slides down a rough incline at constant velocity. This means:

    • the component of gravity along the incline exceeds the friction force
    • the component of gravity along the incline equals the friction force
    • the normal force equals the friction force acting up the incline
    • the net force on the block points down the incline

    Constant velocity means zero net force, so friction (up the incline) balances mg sin(theta).

Unit 4: Circular Motion & Gravitation (14)

  1. A 0.30 kg ball on a 0.60 m string is swung in a horizontal circle, completing 2.0 revolutions per second. Its speed is:

    • 7.5 m/s
    • 3.8 m/s
    • 1.2 m/s
    • 0.60 m/s

    v = 2 pi r f = 2 pi (0.60)(2.0) = 7.54 m/s.

  2. For that ball (0.30 kg, r = 0.60 m, v = 7.54 m/s), the tension in the string is about:

    • 28 N
    • 14 N
    • 4.5 N
    • 57 N

    $T = m v^2/r = 0.30(56.9)/0.60 = 28.4 N$.

  3. A satellite orbits at radius r with speed v. To orbit at radius 4r, its speed must be:

    • v/2
    • 2v
    • v/4
    • 4v

    $v = \sqrt{GM/r}$, so quadrupling r halves the orbital speed.

  4. Planet X has twice Earth's mass and twice Earth's radius. Its surface gravity compared with Earth's is:

    • half as strong
    • twice as strong
    • the same
    • four times as strong

    $g = GM/R^2$; with M x2 and $R^2 x4, g$ is 2/4 = 1/2 of Earth's.

  5. A car goes over a round hill of radius 40 m. The maximum speed at which it stays on the road is $(g = 10 m/s^2)$:

    • 20 m/s
    • 10 m/s
    • 40 m/s
    • 6.3 m/s

    At the limit, gravity supplies all the centripetal force: $v = \sqrt{g r} = \sqrt{400} = 20 m/s$.

  6. The gravitational force between two 100 kg masses 2.0 m apart is about (G = 6.67e-11):

    • 1.7e-7 N
    • 3.3e-7 N
    • 6.7e-9 N
    • 1.3e-6 N

    $F = 6.67e-11 (100 x 100)/(2.0^2) = 6.67e-11 (10000/4) = 1.67e-7 N$.

  7. A ball on a string is swung in a vertical circle of radius 1.0 m. The minimum speed at the top to keep the string taut is $(g = 10 m/s^2)$:

    • 3.2 m/s
    • 10 m/s
    • 1.0 m/s
    • 6.3 m/s

    At minimum, tension = 0 and gravity is the centripetal force: $v = \sqrt{g r} = \sqrt{10} = 3.16 m/s$.

  8. A 2.0 kg mass moves in a circle of radius 0.50 m at 4.0 m/s. The net force on it is:

    • 64 N toward the center
    • 64 N tangent to the circle
    • 16 N toward the center
    • 32 N away from the center

    $F_c = m v^2/r = 2.0(16)/0.50 = 64 N$, directed toward the center.

  9. If Earth were compressed to half its radius with the same mass, surface gravity would become:

    • 4 times as strong
    • 2 times as strong
    • half as strong
    • unchanged

    $g = GM/R^2$; halving R quarters $R^2$, so g increases by 4.

  10. Doubling both the mass of an orbiting satellite and its speed, at the same radius, changes the required centripetal force by a factor of:

    • 8
    • 4
    • 2
    • 6

    $F_c = m v^2/r$; m x2 and $v^2 x4$ gives x8.

  11. The Moon stays in orbit around Earth because:

    • no force acts on it, so it coasts in a circle
    • gravity provides the centripetal force that curves its path
    • it is far enough away to be beyond Earth's gravity
    • the Sun's pull pushes it along a curved path around Earth

    Earth's gravitational pull continuously accelerates the Moon toward Earth, bending its path into an orbit.

  12. Two satellites orbit Earth; satellite A is at radius r and B at radius 2r. Compared to A, satellite B has:

    • a longer period and slower speed
    • a shorter period and faster speed
    • the same period
    • a longer period and faster speed

    Larger orbits are slower $(v = \sqrt{GM/r})$ and have longer periods ($T^2$ proportional to $r^3$).

  13. A 1200 kg car rounds a banked curve designed for no friction at 20 m/s, radius 80 m. The centripetal force on the car is $(g = 10 m/s^2)$:

    • 6000 N
    • 3000 N
    • 12000 N
    • 1500 N

    $F_c = m v^2/r = 1200(400)/80 = 6000 N$.

  14. An object's weight at Earth's surface is 800 N. At an altitude where it is two Earth-radii from Earth's center, its weight is:

    • 200 N
    • 400 N
    • 800 N
    • 100 N

    Weight follows $1/r^2$; at 2 radii the factor is 1/4, so 800/4 = 200 N.

Unit 5: Rotational Motion & Torque (12)

  1. A 0.40 m wrench is pulled with 50 N of force perpendicular to its handle. The torque about the bolt is:

    • 20 N*m
    • 125 N*m
    • 12.5 N*m
    • 2.0 N*m

    tau = r F = 0.40 x 50 = 20 N*m (force is perpendicular, so sin = 1).

  2. That same 50 N force is applied to the 0.40 m wrench at 30 degrees to the handle. The torque is now (sin30 = 0.50):

    • 10 N*m
    • 20 N*m
    • 17 N*m
    • 25 N*m

    tau = r F sin(theta) = 0.40 x 50 x 0.50 = 10 N*m.

  3. A uniform 4.0 m, 20 kg beam is supported at its two ends. Each support carries $(g = 10 m/s^2)$:

    • 100 N
    • 200 N
    • 50 N
    • 400 N

    By symmetry each support holds half the 200 N weight: 100 N.

  4. A seesaw balances with a 30 kg child 2.0 m from the pivot. A 40 kg child must sit how far from the pivot on the other side?

    • 1.5 m
    • 2.7 m
    • 1.0 m
    • 2.0 m

    30(2.0) = 40 d -> d = 60/40 = 1.5 m.

  5. A solid disk $(I = 0.5 M R^2)$ of mass 2.0 kg and radius 0.30 m has a net torque of 0.90 N*m applied. Its angular acceleration is:

    • $10 rad/s^2$
    • $5.0 rad/s^2$
    • $0.10 rad/s^2$
    • $20 rad/s^2$

    $I = 0.5(2.0)(0.09) = 0.09 kg*m^2$; alpha = $tau/I = 0.90/0.09 = 10 rad/s^2$.

  6. A spinning skater with $I = 4.0 kg*m^2$ rotating at 3.0 rad/s pulls in her arms, reducing I to $2.0 kg*m^2$. Her new angular speed is:

    • 6.0 rad/s
    • 1.5 rad/s
    • 3.0 rad/s
    • 12 rad/s

    Angular momentum conserved: 4.0(3.0) = 2.0(omega) -> omega = 6.0 rad/s.

  7. For that skater, her rotational kinetic energy after pulling in her arms compared with before:

    • doubles
    • halves
    • stays the same
    • quadruples

    KE = $0.5 I omega^2$: before 0.5(4.0)(9) = 18 J; after 0.5(2.0)(36) = 36 J. It doubles (her muscles do work).

  8. A wheel starting from rest reaches 20 rad/s after 4.0 s of constant angular acceleration. Through how many radians did it turn?

    • 40 rad
    • 80 rad
    • 20 rad
    • 10 rad

    theta = $0.5(omega_i + omega_f)t = 0.5(0 + 20)(4.0) = 40$ rad.

  9. A point on the rim of a 0.50 m radius wheel spinning at 10 rad/s has a linear speed of:

    • 5.0 m/s
    • 20 m/s
    • 10 m/s
    • 0.05 m/s

    v = r omega = 0.50 x 10 = 5.0 m/s.

  10. A solid sphere and a hollow sphere of equal mass and radius roll down a ramp. Which reaches the bottom first?

    • The solid sphere
    • The hollow sphere
    • They tie
    • The heavier one

    The solid sphere has a smaller $I/(MR^2)$, so less energy goes into rotation and more into translation.

  11. A uniform 6.0 m, 30 kg plank rests on two supports: one at the far left end and one 4.0 m from that end. What force does the right support push up with $(g = 10 m/s^2)?$

    • 225 N
    • 150 N
    • 300 N
    • 75 N

    Take torques about the left support: the 300 N weight acts at the 3.0 m center, so $R_{right}(4.0) = 300(3.0)$ -> $R_{right} = 225 N$.

  12. To loosen a stuck bolt, the most effective action is to:

    • use a longer wrench
    • push harder toward the bolt center
    • use a shorter wrench
    • tap the wrench lightly

    A longer lever arm multiplies the torque for the same applied force.

Unit 6: Momentum & Energy (15)

  1. A 3.0 kg block slides 4.0 m along a rough floor, slowing from 6.0 m/s to 2.0 m/s. How much work did friction do?

    • -48 J
    • -24 J
    • 48 J
    • -16 J

    $W_{net} =$ dKE = $0.5(3.0)(2^2 - 6^2) = 0.5(3.0)(-32) = -48 J$.

  2. From that result, the friction force on the 3.0 kg block over 4.0 m was:

    • 6.0 N
    • 12 N
    • 48 N
    • 3.0 N

    |W| = f d -> f = 48/4.0 = 12 N.

  3. A 0.20 kg ball falls from 10 m and rebounds to $6.0 m (g = 10 m/s^2)$. How much mechanical energy was lost in the bounce?

    • 8.0 J
    • 4.0 J
    • 20 J
    • 12 J

    Lost = mg(h1 - h2) = 0.20(10)(10 - 6.0) = 8.0 J.

  4. A 2.0 kg cart moving at 3.0 m/s collides and sticks to a 4.0 kg cart at rest. Their common final speed is:

    • 1.0 m/s
    • 1.5 m/s
    • 2.0 m/s
    • 0.5 m/s

    p: 2.0(3.0) = 6.0 = (6.0)v -> v = 1.0 m/s.

  5. In that perfectly inelastic collision (2.0 kg at 3.0 m/s into 4.0 kg at rest), how much kinetic energy is lost?

    • 3.0 J
    • 6.0 J
    • 9.0 J
    • 0 J

    KE before = 0.5(2.0)(9) = 9.0 J; KE after = $0.5(6.0)(1.0)^2 = 3.0 J$; lost = 6.0 J.

  6. A spring with k = 400 N/m is compressed 0.10 m and released, launching a 0.50 kg ball horizontally on a frictionless surface. The ball's speed is:

    • 2.8 m/s
    • 4.0 m/s
    • 1.4 m/s
    • 8.0 m/s

    $0.5 k x^2 = 0.5 m v^2$ -> $v = x \sqrt{k/m} = 0.10 \sqrt{800} = 0.10(28.3) = 2.83 m/s$.

  7. A 1500 kg car travels at 20 m/s. The braking force needed to stop it in 50 m is:

    • 6000 N
    • 12000 N
    • 3000 N
    • 30000 N

    KE = 0.5(1500)(400) = 300000 J; F = 300000/50 = 6000 N.

  8. A 60 kg runner climbs a 5.0 m staircase in $4.0 s (g = 10 m/s^2)$. Their average power output against gravity is:

    • 750 W
    • 3000 W
    • 300 W
    • 150 W

    Work = mgh = 60(10)(5.0) = 3000 J; P = 3000/4.0 = 750 W.

  9. A 1000 kg car moves at a steady 25 m/s against 800 N of resistive force. The engine's power output is:

    • 20000 W
    • 25000 W
    • 800 W
    • 31250 W

    At constant speed the driving force equals 800 N; P = Fv = 800 x 25 = 20000 W.

  10. A pendulum bob is released from a height of 0.80 m above its lowest point $(g = 10 m/s^2)$. Its speed at the bottom is:

    • 4.0 m/s
    • 2.8 m/s
    • 8.0 m/s
    • 1.6 m/s

    mgh = $0.5 m v^2$ -> $v = \sqrt{2 x 10 x 0.80} = \sqrt{16} = 4.0 m/s$.

  11. Two identical 1.0 kg carts approach each other at 3.0 m/s and 1.0 m/s and undergo a perfectly inelastic collision. Their final velocity is:

    • 1.0 m/s in the direction of the faster cart
    • 2.0 m/s in the direction of the faster cart
    • zero, since the two momenta cancel completely
    • 1.0 m/s in the direction of the slower cart

    p = 1.0(3.0) + 1.0(-1.0) = 2.0 kg*m/s; v = 2.0/2.0 = 1.0 m/s toward the faster cart's original motion.

  12. A 0.50 kg ball moving right at 4.0 m/s bounces elastically off a wall and moves left at 4.0 m/s. The impulse delivered by the wall is:

    • 4.0 N*s left
    • 2.0 N*s left
    • 0 N*s
    • 8.0 N*s right

    J = dp = 0.50(-4.0 - 4.0) = -4.0 N*s, i.e. 4.0 N*s to the left.

  13. A crane lifts a 500 kg load at a constant $0.20 m/s (g = 10 m/s^2)$. The crane's power is:

    • 1000 W
    • 100 W
    • 5000 W
    • 2000 W

    Force = mg = 5000 N; P = Fv = 5000 x 0.20 = 1000 W.

  14. A block slides down a frictionless ramp from rest at height h. If h is doubled, the speed at the bottom increases by a factor of:

    • 2
    • sqrt(2)
    • 4
    • 1.5

    $v = \sqrt{2gh}$, so doubling h multiplies v by sqrt(2).

  15. A 2.0 kg object has 36 J of kinetic energy. Its speed is:

    • 6.0 m/s
    • 18 m/s
    • 3.0 m/s
    • 9.0 m/s

    $36 = 0.5(2.0)v^2$ -> $v^2 = 36$ -> v = 6.0 m/s.

Unit 7: Electricity (12)

  1. Three 6.0 ohm resistors are connected in parallel. The equivalent resistance is:

    • 2.0 ohms
    • 18 ohms
    • 6.0 ohms
    • 0.5 ohms

    1/R = 3(1/6) = 1/2, so R = 2.0 ohms.

  2. A 12 V battery is connected to a 4.0 ohm and 2.0 ohm resistor in series. The current is:

    • 2.0 A
    • 6.0 A
    • 1.0 A
    • 3.0 A

    $R_{total} = 6.0$ ohms; I = 12/6.0 = 2.0 A.

  3. For that circuit (12 V, 4.0 ohm and 2.0 ohm in series, I = 2.0 A), the voltage across the 4.0 ohm resistor is:

    • 8.0 V
    • 4.0 V
    • 12 V
    • 2.0 V

    V = IR = 2.0 x 4.0 = 8.0 V.

  4. A 3.0 ohm and 6.0 ohm resistor are in parallel across a 12 V battery. The total current from the battery is:

    • 6.0 A
    • 2.0 A
    • 4.0 A
    • 18 A

    $R_{parallel} = (3 x 6)/(3 + 6) = 2.0$ ohms; I = 12/2.0 = 6.0 A.

  5. A resistor dissipates 24 W when 3.0 A flows through it. Its resistance is:

    • 2.7 ohms
    • 8.0 ohms
    • 72 ohms
    • 0.13 ohms

    $P = I^2 R$ -> R = 24/9 = 2.67 ohms.

  6. Two point charges of +2 C and -2 C are 1.0 m apart. Replacing one with +4 C changes the force magnitude by a factor of:

    • 2
    • 4
    • 1/2
    • 1

    Force is proportional to the product of the charges; doubling one charge doubles the force.

  7. A wire's length is doubled and its cross-sectional area is halved. Its resistance:

    • increases by a factor of 4
    • doubles
    • halves
    • is unchanged

    R is proportional to L/A; L x2 and A x(1/2) gives x4.

  8. A 60 W bulb operates on 120 V. The current it draws is:

    • 0.50 A
    • 2.0 A
    • 7200 A
    • 0.017 A

    I = P/V = 60/120 = 0.50 A.

  9. In a series circuit with a 2 ohm and 3 ohm resistor, which statement is true?

    • The 3 ohm resistor has the larger voltage drop
    • The 2 ohm resistor has the larger voltage drop
    • Both have equal voltage drops
    • The current is larger in the 2 ohm resistor

    Same current; V = IR, so the larger resistance has the larger voltage drop.

  10. The electric field 0.20 m from a +5.0e-9 C point charge is about (k = 9.0e9):

    • 1130 N/C
    • 113 N/C
    • 2250 N/C
    • 45 N/C

    $E = k Q/r^2 = 9.0e9 (5.0e-9)/(0.04) = 45/0.04 = 1125 N/C$.

  11. A 100 W device runs for 2.0 hours. The energy it uses is:

    • 720000 J
    • 200 J
    • 50 J
    • 7200 J

    E = P t = 100 W x 7200 s = 720000 J (0.2 kWh).

  12. Adding a second identical bulb in parallel to a battery (negligible internal resistance) makes the original bulb:

    • glow about the same as before
    • glow dimmer
    • glow brighter
    • turn off

    Parallel branches keep the same voltage, so the first bulb's current and brightness are essentially unchanged.

Unit 8: Magnetism & Electromagnetism (12)

  1. A proton moves at 2.0e5 m/s perpendicular to a 0.50 T magnetic field. The magnetic force on it is (q = 1.6e-19 C):

    • 1.6e-14 N
    • 3.2e-14 N
    • 8.0e-14 N
    • 1.6e-19 N

    F = qvB = 1.6e-19 x 2.0e5 x 0.50 = 1.6e-14 N.

  2. An electron enters a uniform magnetic field moving perpendicular to it. Its path is:

    • a straight line
    • a circle
    • a parabola
    • a spiral that speeds up

    A constant perpendicular magnetic force of fixed magnitude produces uniform circular motion.

  3. A 0.20 m wire carries 3.0 A perpendicular to a 0.40 T field. The force on the wire is:

    • 0.24 N
    • 2.4 N
    • 0.024 N
    • 1.2 N

    F = B I L = 0.40 x 3.0 x 0.20 = 0.24 N.

  4. A coil of 50 turns has its flux change from 0.020 Wb to 0.060 Wb in 0.10 s. The average induced EMF magnitude is:

    • 20 V
    • 0.40 V
    • 2.0 V
    • 200 V

    EMF = N dPhi/dt = 50 (0.040/0.10) = 50 x 0.40 = 20 V.

  5. An ideal transformer steps 120 V down to 12 V. If the primary has 500 turns, the secondary has:

    • 50 turns
    • 5000 turns
    • 250 turns
    • 100 turns

    $N_s = N_p (V_s/V_p) = 500 (12/120) = 50$ turns.

  6. That transformer (120 V to 12 V) delivers 2.0 A to the load. The ideal primary current is:

    • 0.20 A
    • 2.0 A
    • 20 A
    • 0.02 A

    Power in = power out: $120 I_p = 12 x 2.0$ -> $I_p = 24/120 = 0.20 A$.

  7. A bar magnet is pushed north-pole-first toward a coil. The induced current in the coil creates a magnetic pole facing the magnet that is:

    • south, to attract the approaching magnet
    • north, to repel the approaching magnet
    • alternating, switching poles at the magnet's speed
    • zero, since the coil is stationary

    By Lenz's law the induced current opposes the approach, presenting a like (north) pole.

  8. A charged particle passes undeflected through crossed electric and magnetic fields (a velocity selector). Its speed is:

    • E/B
    • B/E
    • E B
    • E + B

    Balance requires qE = qvB, so v = E/B.

  9. Doubling the speed of a charged particle in a fixed magnetic field (perpendicular entry) changes the radius of its circular path by a factor of:

    • 2
    • 1/2
    • 4
    • 1

    r = mv/(qB), so radius is proportional to speed.

  10. A loop of wire lies flat in a magnetic field that is increasing in strength. The induced current:

    • flows to oppose the increase in flux
    • flows to increase the flux
    • is zero because the loop does not move
    • reverses every instant

    Lenz's law: the induced current opposes the change, here the increasing flux.

  11. A generator's output frequency is doubled by spinning its coil twice as fast. The peak induced EMF:

    • also doubles
    • is unchanged
    • is halved
    • quadruples

    Faster rotation means a faster rate of flux change, so the peak EMF scales up with rotation rate.

  12. Which change will NOT induce a current in a stationary coil?

    • Pulling a strong magnet out of the coil quickly
    • Rotating a nearby magnet beside the coil
    • Switching a nearby electromagnet on near the coil
    • Holding a strong magnet motionless inside the coil

    A motionless magnet gives constant flux; only a changing flux induces current.

Unit 9: Simple Harmonic Motion (12)

  1. A 0.25 kg mass on a spring oscillates with a period of 0.50 s. The spring constant is (use $4 pi^2$ ~ 39.5):

    • about 39.5 N/m
    • about 3.1 N/m
    • about 100 N/m
    • about 10 N/m

    T = 2 pi $\sqrt{m/k}$ -> $k = 4 pi^2 m/T^2 = 39.5(0.25)/0.25 = 39.5 N/m$.

  2. A simple pendulum has a period of 2.0 s. To double the period to 4.0 s, its length must be:

    • multiplied by 4
    • multiplied by 2
    • halved
    • multiplied by sqrt(2)

    T is proportional to sqrt(L); doubling T requires 4x the length.

  3. A 0.10 kg mass on a spring (k = 40 N/m) has amplitude 0.20 m. Its maximum speed is:

    • 4.0 m/s
    • 2.0 m/s
    • 0.80 m/s
    • 8.0 m/s

    $v_{max} = A \sqrt{k/m} = 0.20 \sqrt{400} = 0.20 x 20 = 4.0 m/s$.

  4. For that oscillator (m = 0.10 kg, k = 40 N/m, A = 0.20 m), the total mechanical energy is:

    • 0.80 J
    • 1.6 J
    • 0.40 J
    • 8.0 J

    $E = 0.5 k A^2 = 0.5(40)(0.04) = 0.80 J$.

  5. At what point in its oscillation does a mass on a spring have all kinetic energy and no potential energy?

    • At the equilibrium position
    • At maximum displacement
    • Halfway to the amplitude
    • It never does

    At equilibrium the spring is at natural length (PE = 0) and speed is maximum.

  6. A pendulum on Earth has period T. On a planet where g is 4 times Earth's, its period becomes:

    • T/2
    • 2T
    • 4T
    • T

    T is proportional to $1/\sqrt{g}$; quadrupling g halves the period.

  7. A mass-spring system's amplitude is doubled. Its maximum speed:

    • doubles
    • stays the same
    • halves
    • quadruples

    $v_{max} = A$ omega, and omega does not depend on amplitude, so $v_{max}$ doubles.

  8. A child on a swing is pushed once each cycle at the swing's natural frequency. The amplitude grows. This is:

    • resonance
    • damping
    • interference
    • the Doppler effect

    Driving a system at its natural frequency efficiently transfers energy, building amplitude (resonance).

  9. A lightly damped oscillator loses energy slowly. Over several cycles, its period:

    • increases sharply while amplitude decreases
    • stays nearly constant while amplitude decreases
    • decreases sharply while amplitude stays constant
    • doubles with each successive cycle that passes

    Light damping mainly shrinks amplitude; the period changes very little.

  10. The acceleration of a SHM oscillator is zero at the moment its:

    • displacement is zero (equilibrium)
    • displacement is maximum
    • speed is zero
    • kinetic energy is zero

    Acceleration is proportional to -x, so it is zero exactly where x = 0.

  11. Two mass-spring systems have the same spring but masses m and 4m. The ratio of their periods (heavier to lighter) is:

    • 2 to 1
    • 4 to 1
    • 1 to 2
    • 1 to 1

    T is proportional to sqrt(m); $\sqrt{4m/m} = 2$.

  12. A 1.0 m simple pendulum is timed for 10 complete swings, taking 20 s. The measured g is closest to:

    • $9.9 m/s^2$
    • $4.9 m/s^2$
    • $19.7 m/s^2$
    • $2.0 m/s^2$

    T = 2.0 s; $g = 4 pi^2 L/T^2 = 39.5(1.0)/4.0 = 9.87 m/s^2$.

Unit 10: Waves & Sound (12)

  1. A wave travels 240 m in 3.0 s. If its frequency is 40 Hz, its wavelength is:

    • 2.0 m
    • 80 m
    • 0.5 m
    • 6.0 m

    v = 240/3.0 = 80 m/s; lambda = v/f = 80/40 = 2.0 m.

  2. A string 1.2 m long fixed at both ends vibrates in 3 segments (third harmonic). The wavelength is:

    • 0.80 m
    • 1.2 m
    • 2.4 m
    • 0.40 m

    L = 3(lambda/2) -> lambda = 2L/3 = 2.4/3 = 0.80 m.

  3. For that string (L = 1.2 m, third harmonic, wave speed 120 m/s), the frequency is:

    • 150 Hz
    • 100 Hz
    • 50 Hz
    • 300 Hz

    lambda = 0.80 m; f = v/lambda = 120/0.80 = 150 Hz.

  4. A car horn emits 400 Hz. As the car speeds toward a stationary listener, the listener hears a frequency that is:

    • higher than 400 Hz
    • lower than 400 Hz
    • exactly 400 Hz
    • zero

    Approaching source compresses wavefronts, raising the observed frequency.

  5. Sound travels at about 340 m/s in air. A person hears an echo from a cliff 1.0 s after shouting. The cliff is:

    • 170 m away
    • 340 m away
    • 680 m away
    • 85 m away

    Sound covers there and back: 340 m total, so the cliff is 170 m away.

  6. Two waves of equal amplitude A meet exactly out of phase. The resulting amplitude is:

    • 0
    • A
    • 2A
    • A/2

    Completely destructive interference: the displacements cancel.

  7. A wave's frequency is tripled in the same medium. Its speed:

    • triples
    • is unchanged
    • is one-third
    • is nine times as large

    Wave speed is fixed by the medium; wavelength drops to one-third instead.

  8. Middle C is 262 Hz. Its first overtone (second harmonic) on a both-ends-fixed string is:

    • 524 Hz
    • 131 Hz
    • 786 Hz
    • 262 Hz

    Harmonics are integer multiples of the fundamental: 2 x 262 = 524 Hz.

  9. A tuning fork of 512 Hz produces 3 beats per second with a guitar string. The string's frequency is:

    • 509 Hz or 515 Hz
    • 512 Hz
    • 256 Hz or 1024 Hz
    • 3 Hz

    Beat frequency is the difference, so the string is 512 +/- 3 Hz.

  10. A wave slows down entering a new medium. Its wavelength:

    • decreases
    • increases
    • stays the same
    • becomes zero

    Frequency is unchanged; since v = f lambda and v drops, lambda decreases.

  11. An ambulance siren sounds 700 Hz while approaching and 620 Hz after passing. The true source frequency is:

    • around 660 Hz
    • 700 Hz
    • 620 Hz
    • 1320 Hz

    The actual frequency is between the approaching (higher) and receding (lower) values.

  12. Doubling the amplitude of a wave changes the energy it carries by a factor of about:

    • 4
    • 2
    • 1
    • 8

    Wave energy is proportional to amplitude squared, so doubling amplitude roughly quadruples the energy.

Unit 11: Optics (12)

  1. Light travels from air (n = 1.0) into glass (n = 1.5) at a 30 degree angle of incidence. The refraction angle is (sin30 = 0.50):

    • about 19 degrees
    • about 30 degrees
    • about 49 degrees
    • about 45 degrees

    1.0 sin30 = 1.5 sin(theta) -> sin(theta) = 0.333 -> theta = 19.5 degrees.

  2. The speed of light in a medium with n = 2.0 is:

    • 1.5e8 m/s
    • 3.0e8 m/s
    • 6.0e8 m/s
    • 2.0e8 m/s

    v = c/n = 3.0e8/2.0 = 1.5e8 m/s.

  3. An object is placed 30 cm from a converging lens of focal length 10 cm. The image distance is:

    • 15 cm
    • 30 cm
    • 7.5 cm
    • -15 cm

    1/10 = 1/30 + 1/di -> 1/di = 1/10 - 1/30 = 2/30 -> di = 15 cm.

  4. For that lens image (do = 30 cm, di = 15 cm), the magnification is:

    • -0.5 (inverted, reduced)
    • +0.5 (upright, reduced)
    • -2 (inverted, enlarged)
    • +2 (upright, enlarged)

    m = -di/do = -15/30 = -0.5; negative means inverted, magnitude < 1 means reduced.

  5. The critical angle for a glass-air boundary where $n_{glass} = 1.5$ is $(\sin(theta_c) = 1/1.5)$:

    • about 42 degrees
    • about 30 degrees
    • about 48 degrees
    • about 60 degrees

    $\sin(theta_c) = 1/1.5 = 0.667$ -> $theta_c = 41.8$ degrees.

  6. A concave mirror has a focal length of 12 cm. An object 12 cm away forms an image:

    • at infinity (no image forms)
    • 12 cm in front, same size
    • 6 cm in front, inverted
    • 24 cm in front, inverted

    With do = f, 1/di = 1/f - 1/do = 0, so the reflected rays are parallel and no image forms.

  7. Red light (700 nm) and blue light (450 nm) enter a prism. Which bends more?

    • Red, because its longer wavelength gives a higher index
    • They bend equally, because both travel at c in glass
    • Blue, because the index is slightly higher for shorter wavelengths
    • Neither, because the prism's faces are flat and not curved

    Dispersion: the refractive index is slightly larger for shorter (blue) wavelengths, so blue bends more.

  8. A radio wave has a frequency of 1.0e8 Hz. Its wavelength in vacuum is:

    • 3.0 m
    • 0.33 m
    • 3.0e16 m
    • 30 m

    lambda = c/f = 3.0e8/1.0e8 = 3.0 m.

  9. When light goes from water (n = 1.33) into air, it bends:

    • away from the normal
    • toward the normal
    • straight through with no bend
    • back into the water always

    Going to a less dense (lower n) medium, light speeds up and bends away from the normal (until TIR).

  10. An object is placed 5 cm from a converging lens with f = 10 cm. The image is:

    • real, inverted, and enlarged (a projector)
    • virtual, upright, and enlarged (a magnifying glass)
    • real, inverted, and reduced (a camera)
    • virtual, upright, and reduced (a peephole)

    With the object inside the focal length, a converging lens forms a virtual, upright, magnified image.

  11. Two coherent light sources 0.10 mm apart produce fringes on a screen. Doubling the wavelength used will:

    • double the fringe spacing
    • halve the fringe spacing
    • not change the spacing
    • eliminate the fringes

    Fringe spacing is proportional to wavelength, so doubling lambda doubles the spacing.

  12. A light ray hits a plane mirror at 25 degrees from the mirror surface. The angle of reflection (from the normal) is:

    • 65 degrees
    • 25 degrees
    • 50 degrees
    • 90 degrees

    Angle from the normal = 90 - 25 = 65 degrees; reflection equals incidence.

Unit 12: Modern Physics (11)

  1. A photon has a frequency of 5.0e14 Hz. Its energy is (h = 6.63e-34 J*s):

    • 3.3e-19 J
    • 1.3e-48 J
    • 6.6e-34 J
    • 7.5e47 J

    E = hf = 6.63e-34 x 5.0e14 = 3.3e-19 J.

  2. A metal has a work function of 2.0 eV. Light of photon energy 3.5 eV strikes it. The maximum kinetic energy of ejected electrons is:

    • 1.5 eV
    • 5.5 eV
    • 2.0 eV
    • 3.5 eV

    $KE_{max} =$ hf - W = 3.5 - 2.0 = 1.5 eV.

  3. For that metal (work function 2.0 eV), light of 1.5 eV photons will:

    • eject no electrons
    • eject electrons with 0.5 eV kinetic energy
    • eject electrons only if very bright
    • eject electrons with 3.5 eV kinetic energy

    Photon energy is below the work function, so no electrons are ejected regardless of intensity.

  4. An electron drops from a -3.4 eV level to a -13.6 eV level in hydrogen. The emitted photon energy is:

    • 10.2 eV
    • 17.0 eV
    • 3.4 eV
    • 13.6 eV

    $E_{photon} = E_{high} - E_{low} = (-3.4) - (-13.6) = 10.2$ eV.

  5. A sample has a half-life of 8.0 days. After 24 days, the fraction of the original nuclei remaining is:

    • 1/8
    • 1/3
    • 1/16
    • 1/24

    24 days is 3 half-lives: $(1/2)^3 = 1/8$ remains.

  6. Uranium-238 (92 protons) undergoes alpha decay. The daughter nucleus has:

    • 90 protons and mass number 234
    • 94 protons and mass number 242
    • 92 protons and mass number 234
    • 91 protons and mass number 237

    Alpha decay removes 2 protons and 4 nucleons: Z = 90, A = 234 (thorium-234).

  7. A nucleus emits a beta-minus particle. Its atomic number:

    • increases by 1
    • decreases by 1
    • stays the same
    • decreases by 2

    A neutron becomes a proton and an electron is emitted, so Z rises by 1 while A is unchanged.

  8. Converting 1.0 gram of mass entirely to energy releases about (c = 3.0e8 m/s):

    • 9.0e13 J
    • 3.0e5 J
    • 9.0e16 J
    • 3.0e8 J

    $E = mc^2 = 0.001 x (3.0e8)^2 = 0.001 x 9.0e16 = 9.0e13 J$.

  9. Doubling the frequency of light used in a photoelectric experiment (already above threshold) will:

    • double the photon energy and increase electron KE
    • double the number of electrons ejected each second
    • halve the stopping voltage of the electrons
    • double the work function of the metal

    E = hf, so doubling f doubles each photon's energy and raises $KE_{max} =$ hf - W.

  10. Which process releases energy by joining light nuclei into a heavier one?

    • Nuclear fusion
    • Nuclear fission
    • Alpha decay
    • Ionization

    Fusion combines light nuclei (e.g. hydrogen isotopes) and powers stars.

  11. An isotope's activity drops to 1/4 of its initial value in 30 minutes. Its half-life is:

    • 15 minutes
    • 30 minutes
    • 60 minutes
    • 7.5 minutes

    $1/4 = (1/2)^2$, so 30 minutes equals 2 half-lives, giving 15 minutes each.

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Unit 1: Measurement, Vectors & Motion Basics

Base vs. derived unit
A base unit (m, kg, s, A, K, mol, cd) is defined on its own; a derived unit is a combination of base units, such as m/s for speed or N = kg·m/s².
SI unit of mass
The kilogram (kg), not the gram.
Scientific notation
M × 10^n with 1 ≤ M < 10 and n an integer. Add/subtract with matching powers of 10; multiply → add exponents; divide → subtract exponents.
Order of magnitude
The power of 10 closest to a value: 1284 kg → 10^3 kg, but 8756 kg → 10^4 kg.
Metric prefixes
tera 10^12, giga 10^9, mega 10^6, kilo 10^3, centi 10^-2, milli 10^-3, micro 10^-6, nano 10^-9, pico 10^-12.
Precision vs. accuracy
Precision is the smallest decimal place a measurement reports; accuracy is how close it is to the accepted value.
Systematic vs. random error
Systematic errors push all readings the same way (calibrate to fix); random errors scatter readings (average many trials).
Percent error
|measured − accepted| / accepted × 100%.
Scalar
A quantity with magnitude only: mass, distance, speed, time, energy, temperature.
Vector
A quantity with magnitude and direction: displacement, velocity, acceleration, force, momentum.
Resultant
The single vector equal to the sum of two or more vectors; found head-to-tail or by the parallelogram method.
Range of a resultant
For vectors A and B: |A − B| ≤ R ≤ A + B; the maximum is at 0° between them and the minimum at 180°.
Perpendicular vectors
R = √(A² + B²), direction from tanθ = B/A.
Vector components
A_x = A cosθ and A_y = A sinθ, with θ measured from the horizontal.
Equilibrant
The vector with the same magnitude as the resultant but the opposite direction; it makes the net vector zero.
Distance vs. displacement
Distance is total path length (scalar); displacement is the straight-line change in position with direction (vector).
Average speed vs. average velocity
Speed = total distance / total time (scalar); velocity = displacement / time (vector).
Instantaneous velocity
Speed and direction at a single instant, like a speedometer reading plus a compass heading. On a position-time graph it is the slope of the tangent line at that instant.
Uniform motion
Motion at constant velocity: the same speed in the same direction. Horizontal line on a v–t graph; straight sloped line on a d–t graph.
Slope of a distance–time graph
The velocity. Its size is the speed; a positive slope means motion in the positive direction, a negative slope the negative direction.
Area under a velocity–time graph
The distance traveled. Split it into rectangles and triangles to compute it.
Acceleration
The rate of change of velocity, a = Δv/Δt, in m/s². A vector: it can come from a change in speed, direction, or both.
Deceleration
Acceleration opposite to the velocity, so the object slows down. Not the same as 'negative acceleration', which only describes direction.
Instantaneous vs. average velocity (on a d–t curve)
Instantaneous: slope of the tangent at one point. Average: slope of the straight line joining the start and end points.
Order-of-magnitude benchmarks
30-story building ≈ 10² m; football field ≈ 100 m; little finger ≈ 0.01 m wide; pencil ≈ 5 × 10⁻³ kg; adult ≈ 70 kg.
Vector scale
A fixed ratio between arrow length and magnitude, e.g. 1 mm = 2 m/s. If 15 mm = 30 m/s, then 20 m/s = 10 mm.
Relative velocity
Velocities add as vectors: 4.0 m/s forward on a 2.0 m/s bus is 6.0 m/s relative to the street; toward the back, 2.0 m/s the other way.
Average speed of a round trip
Total distance ÷ total time, not the average of the two speeds (150 km at 60 km/h and back at 75 km/h ≈ 67 km/h).
SOH-CAH-TOA
sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent.
Volume of a can (cylinder)
V = πr²h, using the radius (half the diameter); 1 cm³ = 1 mL.
Slope of a velocity–time graph
The acceleration. Positive slope: velocity increasing. Negative slope: velocity decreasing. Zero slope (a flat line): no acceleration, so constant velocity.

Unit 2: Kinematics

Displacement
Vector change in position, Δx = xf − xi. Can be zero even after a long trip if you return to start.
Distance
Scalar total path length traveled; always ≥ magnitude of displacement.
Free fall acceleration (g)
9.8 m/s² downward near Earth's surface, same for all masses (ignoring air resistance).
Kinematic equation (no x term)
vf = vi + at
Kinematic equation (no vf term)
Δx = vi·t + ½at²
Kinematic equation (no t term)
vf² = vi² + 2aΔx
Projectile motion
Horizontal velocity stays constant; vertical velocity changes due to gravity — the two directions are independent.
Projectile: independence of motion
Horizontal velocity is constant (a_x = 0); vertical motion is free fall (a_y = -g). The two axes share only the time t.
Range of a projectile (level ground)
R = v0^2 sin(2*theta) / g. Maximum at theta = 45 degrees; complementary angles give equal range.
Time of flight (level ground)
t = 2 v0 sin(theta) / g, which is twice the time to reach maximum height.
Free fall
Motion with gravity as the only force. All objects fall with the same acceleration, g ≈ 9.8 m/s² downward (Galileo, not Aristotle).
Reaction time from a dropped ruler
t = √(2d/g), where d is how far the ruler fell before you caught it.

Unit 3: Newton's Laws & Dynamics

Newton's First Law
An object at rest or in constant-velocity motion stays that way unless acted on by a net external force (inertia).
Newton's Second Law
Fnet = ma — acceleration is proportional to net force and inversely proportional to mass.
Newton's Third Law
For every action force, there's an equal and opposite reaction force acting on a DIFFERENT object.
Normal force
Contact force perpendicular to a surface, preventing objects from passing through each other.
Static friction
Friction that resists the START of sliding; maximum value fs,max = μs·FN.
Kinetic friction
Friction that resists sliding once motion has begun: fk = μk·FN.
Free-body diagram
A diagram isolating one object, showing every external force acting on it as an arrow.
Weight
The gravitational force on an object, Fg = mg — not the same as mass.
Normal force on an incline
N = mg cos(theta); the gravity component along the incline is mg sin(theta).
Apparent weight in an elevator
N = m(g + a) accelerating up, N = m(g - a) accelerating down; N = 0 in free fall.

Unit 4: Circular Motion & Gravitation

Centripetal acceleration
ac = v²/r, always directed toward the center of a circular path.
Centripetal force
Fc = mv²/r — the net inward force needed for circular motion, supplied by tension, gravity, friction, or normal force.
Universal gravitation
Fg = Gm1m2/r² — attractive force between any two masses, inverse-square with distance.
Kepler's Third Law
T² ∝ r³ — orbital period squared is proportional to orbital radius cubed.
Orbital speed
v = √(Gm1/r) — depends on the central mass and orbital radius, not the satellite's own mass.
Circular motion speed
v = 2πr/T — speed of an object moving in a circle of radius r with period T.
Kepler's First Law
Planets orbit the Sun in ellipses, with the Sun at one focus.
Kepler's Second Law
A line from the Sun to a planet sweeps out equal areas in equal times — planets move faster near the Sun.
Gravitational constant G
6.67×10⁻¹¹ N·m²/kg² — the proportionality constant in Newton's law of universal gravitation.
Why astronauts feel weightless
The spacecraft and astronaut are in free fall together, so there is no normal force between them; gravity is still present.

Unit 5: Rotational Motion & Torque

Torque
tau = r F sin(theta) = (perpendicular lever arm) x force, in N*m. A force through the axis gives zero torque.
Rotational Newton's second law
Net torque = I * alpha, where I is the moment of inertia and alpha is the angular acceleration.
Moment of inertia
Rotational 'mass'; depends on mass and its distance from the axis (proportional to r^2). Hoop MR^2, disk 0.5 MR^2, solid sphere 0.4 MR^2.
Angular momentum
L = I omega. Conserved when no external torque acts: I1 omega1 = I2 omega2 (skater pulling in arms spins faster).
Rotational kinematics
Same equations as linear motion with x -> theta, v -> omega, a -> alpha. Also s = r theta, v = r omega, a_t = r alpha.
Rotational equilibrium
Sum of torques = 0 and sum of forces = 0. Torques may be taken about any convenient point.
Rolling energy
A rolling object has KE = 0.5 m v^2 + 0.5 I omega^2, so it accelerates down a ramp slower than a sliding one.

Unit 6: Momentum & Energy

Momentum
p = mv, a vector describing an object's quantity of motion.
Impulse
J = FΔt = Δp — the change in momentum caused by a force acting over time.
Conservation of momentum
In an isolated system, total momentum before an interaction equals total momentum after.
Elastic collision
Momentum AND kinetic energy are both conserved; objects bounce apart without permanent deformation.
Inelastic collision
Momentum is conserved, but kinetic energy is not — some converts to heat/sound/deformation.
Work
W = Fd cosθ — energy transferred by a force acting through a displacement.
Work-energy theorem
The net work done on an object equals its change in kinetic energy: Wnet = ΔKE.
Kinetic energy
KE = ½mv² — energy of motion, always non-negative.
Conservation of mechanical energy
KEi + PEi = KEf + PEf when only conservative forces (gravity, springs) act on the system.
Impulse-momentum theorem
Impulse J = F_avg * delta t = change in momentum delta p. Units: N*s = kg*m/s.
Perfectly inelastic collision
Objects stick together: m1 v1 + m2 v2 = (m1 + m2) v_f. Momentum conserved, kinetic energy is not.
Power
P = W/t = energy/time (watts). Also P = F v for a force moving an object at speed v.

Unit 7: Electricity

Coulomb's Law
Fe = kq1q2/r² — the electric force between two point charges, inverse-square with distance.
Electric field
E = F/q — force per unit positive test charge; points away from + charges, toward − charges.
Ohm's Law
V = IR — relates voltage, current, and resistance.
Series circuit
Same current everywhere; total resistance = sum of individual resistances.
Parallel circuit
Same voltage across every branch; total resistance is less than the smallest individual resistor.
Electric power
P = IV = I²R = V²/R — rate of electrical energy transfer.
Charge quantization
Charge is quantized: q = ne, where e = 1.6×10⁻¹⁹ C is the elementary charge.
Conservation of charge
Total charge in an isolated system is constant — charge is transferred, never created or destroyed.
Resistivity
A material property in R = ρL/A; resistance increases with length, decreases with cross-sectional area.
Series vs parallel resistance
Series: R_total = R1 + R2 + ... (increases). Parallel: 1/R_total = 1/R1 + 1/R2 + ... (decreases, less than the smallest).
Electric power (three forms)
P = I V = I^2 R = V^2 / R, measured in watts.

Unit 8: Magnetism & Electromagnetism

Magnetic force on a moving charge
F = qvB sinθ — zero if velocity is parallel to the field, maximum if perpendicular.
Right-hand rule
Used to find the direction of a magnetic field around a current, or the force on a moving charge in a field.
Faraday's Law
A changing magnetic flux through a loop induces an EMF (voltage) in that loop.
Lenz's Law
Induced current flows in a direction that opposes the change in flux that created it.
Force on a current-carrying wire
F = BIL sinθ — the magnetic force on a wire of length L carrying current I in field B.
Motor
Converts electrical energy to mechanical energy using the force on a current-carrying coil in a magnetic field.
Generator
Converts mechanical energy to electrical energy by rotating a coil within a magnetic field, inducing an EMF.
Force on a moving charge
F = q v B sin(theta), perpendicular to both v and B. Zero when v is parallel to B; maximum when perpendicular.
Ideal transformer
V_s / V_p = N_s / N_p. Ideal power is conserved, so V_p I_p = V_s I_s.

Unit 9: Simple Harmonic Motion

Simple harmonic motion condition
A restoring force proportional to displacement: F = -k x. The period is independent of amplitude.
Mass-spring period
T = 2 pi sqrt(m / k). Stiffer spring or lighter mass gives a shorter period; independent of g and amplitude.
Simple pendulum period
T = 2 pi sqrt(L / g) for small angles. Depends only on length and g, not on mass or amplitude.
Energy in SHM
Total energy = 0.5 k A^2. All potential energy at the turning points, all kinetic energy at equilibrium.
Maximum speed in SHM
v_max = A sqrt(k/m) = A omega, occurring as the oscillator passes through equilibrium.
Where acceleration is largest in SHM
At the turning points (maximum displacement), because acceleration is proportional to -x. It is zero at equilibrium.
Damping
Energy loss (friction, air resistance) that gradually reduces amplitude while barely changing the period.
Resonance
Driving a system at one of its natural frequencies transfers energy efficiently and builds a large amplitude.

Unit 10: Waves & Sound

Wavelength
Distance between two successive identical points on a wave (e.g., crest to crest).
Frequency
Number of complete wave cycles per second, measured in hertz (Hz).
Wave speed equation
v = fλ — determined by the medium, not by frequency or amplitude alone.
Transverse wave
Particles vibrate perpendicular to the direction of energy travel (e.g., light, waves on a string).
Longitudinal wave
Particles vibrate parallel to the direction of energy travel, with compressions and rarefactions (e.g., sound).
Constructive interference
In-phase waves add together, producing a larger amplitude.
Doppler effect
Apparent shift in observed frequency due to relative motion between a wave source and observer.
Standing wave on a string (both ends fixed)
L = n * lambda / 2; harmonic frequencies f_n = n * v / (2L), n = 1, 2, 3, ...

Unit 11: Optics

Law of reflection
Angle of incidence equals angle of reflection, both measured from the normal.
Index of refraction
n = c/v — ratio of light's speed in a vacuum to its speed in a medium; always ≥ 1.
Snell's Law
n1 sinθ1 = n2 sinθ2 — relates the bending of light at a boundary between two media.
Total internal reflection
Occurs when light travels from a denser to less dense medium beyond the critical angle — all light reflects, none refracts out.
Convex (converging) lens
Thicker in the middle; focuses parallel rays to a real focal point.
Thin lens equation
1/f = 1/do + 1/di — relates focal length, object distance, and image distance.

Unit 12: Modern Physics

Photoelectric effect
Light above a threshold frequency ejects electrons from a metal; explained by treating light as photons, E = hf.
Work function
W0 — the minimum energy needed to eject an electron from a metal's surface.
De Broglie wavelength
λ = h/p — matter waves; all moving matter has an associated wavelength.
Half-life
The time for half of a radioactive sample to decay; decay is exponential, not linear.
Alpha decay
Nucleus emits a helium nucleus (2p+2n); mass number −4, atomic number −2; stopped by paper.
Gamma decay
Nucleus emits high-energy EM radiation; no change in mass or atomic number; needs lead/concrete to stop.
Beta decay
A neutron converts to a proton (emitting an electron); mass number unchanged, atomic number ±1; stopped by aluminum.
Nuclear fusion
Combining light nuclei into a heavier one, releasing large amounts of energy — powers the Sun.
Photon energy
E = h f = h c / lambda, with h = 6.63e-34 J*s. Higher frequency means more energy per photon.
Press A–F to answer · Enter for next
Try each problem on your own first — then reveal the solution one step at a time. Mark “Got it” to track your progress.

Unit 1: Measurement, Vectors & Motion Basics

Adding perpendicular displacements
A student walks 30 m east, then 40 m north. Find the magnitude and direction of the resultant displacement.
Percent error of a measurement
A lab group measures free-fall acceleration as 9.62 m/s². The accepted value is 9.80 m/s². What is their percent error?

Unit 2: Kinematics

Constant acceleration from rest
A car starts from rest and accelerates at 3.0 m/s² for 6.0 s. Find its final speed and the distance it covers.
Ball rolling off a table
A ball rolls off a 1.25 m high table at 2.0 m/s horizontally. How long is it in the air, and how far from the table does it land? (g = 9.8 m/s²)

Unit 3: Newton's Laws & Dynamics

Pulling a box against friction
A 5.0 kg box is pulled across a floor by a 30 N horizontal force. The coefficient of kinetic friction is 0.20. Find the box's acceleration. (g = 9.8 m/s²)
Apparent weight in an elevator
A 60 kg student stands on a scale in an elevator accelerating upward at 2.0 m/s². What does the scale read? (g = 9.8 m/s²)

Unit 4: Circular Motion & Gravitation

Friction needed on a flat curve
A 1200 kg car rounds a flat curve of radius 50 m at 15 m/s. What centripetal force is needed, and what is the minimum coefficient of static friction? (g = 9.8 m/s²)
Satellite in circular orbit
A 1000 kg satellite orbits Earth at r = 7.0 × 10⁶ m from Earth's center. Find the gravitational force on it and its orbital speed. (G = 6.67 × 10⁻¹¹ N·m²/kg², M_Earth = 5.97 × 10²⁴ kg)

Unit 5: Rotational Motion & Torque

Torque on a wrench
A 40 N force is applied at the end of a 0.25 m wrench. Find the torque when the force is perpendicular to the wrench, and when it is at 60° to the wrench.
Balancing a seesaw
A 30 kg child sits 2.0 m from the pivot of a seesaw. Where must a 40 kg child sit on the other side to balance it?

Unit 6: Momentum & Energy

Cars that stick together
A 1500 kg car moving at 20 m/s hits a stationary 1000 kg car, and they lock together. Find their speed just after the collision and the kinetic energy lost.
Speed on a frictionless track
A roller coaster starts from rest at the top of a 20 m hill. Ignoring friction, how fast is it moving when it is 5.0 m above the ground? (g = 9.8 m/s²)

Unit 7: Electricity

A series–parallel circuit
A 12 V battery is connected to a 4.0 Ω resistor in series with a parallel pair of 6.0 Ω and 3.0 Ω resistors. Find the total current and the current through each parallel resistor.
Force between two charges
Charges of +3.0 μC and −2.0 μC are 0.10 m apart. Find the magnitude and type of the force between them. (k = 8.99 × 10⁹ N·m²/C²)

Unit 8: Magnetism & Electromagnetism

Proton in a magnetic field
A proton moves at 2.0 × 10⁶ m/s perpendicular to a 0.50 T magnetic field. Find the magnetic force on it and the radius of its circular path. (q = 1.6 × 10⁻¹⁹ C, m = 1.67 × 10⁻²⁷ kg)
Step-down transformer
An ideal transformer has 500 turns on its primary, connected to 120 V, and 25 turns on its secondary. Find the secondary voltage, and the primary current when the secondary delivers 2.0 A.

Unit 9: Simple Harmonic Motion

Mass on a spring
A 0.50 kg mass on a spring with k = 200 N/m oscillates with an amplitude of 0.10 m. Find the period, the frequency and the maximum speed.
Designing a 2-second pendulum
How long must a simple pendulum be to have a period of 2.0 s? (g = 9.8 m/s²)

Unit 10: Waves & Sound

Wavelength of a musical note
The note A4 has a frequency of 440 Hz. If sound travels at 343 m/s in air, what is its wavelength?
Harmonics on a guitar string
A 0.65 m guitar string, fixed at both ends, carries waves at 260 m/s. Find its fundamental frequency and its third-harmonic frequency.

Unit 11: Optics

Refraction into glass
Light in air hits a glass block (n = 1.50) at 40° from the normal. Find the angle of refraction and the speed of light in the glass. (c = 3.0 × 10⁸ m/s)
Image from a converging lens
An object is 30 cm in front of a converging lens with focal length 10 cm. Where is the image, and what are its size and orientation?

Unit 12: Modern Physics

Photoelectric effect with green light
Light of wavelength 500 nm shines on a metal with a work function of 2.0 eV. Find the photon energy and the maximum kinetic energy of the ejected electrons. (h = 6.63 × 10⁻³⁴ J·s, c = 3.0 × 10⁸ m/s, 1 eV = 1.6 × 10⁻¹⁹ J)
Radioactive half-life
A sample contains 80 g of an isotope with a half-life of 5.0 days. How much remains after 20 days?

Core Formulas

Kinematics 1
vf = vi + at
Kinematics 2
Δx = vi·t + ½at²
Kinematics 3
vf² = vi² + 2aΔx
Kinematics 4
Δx = ½(vi+vf)t
Newton's 2nd Law
Fnet = ma
Weight
Fg = mg
Friction
ff = μFN
Momentum
p = mv
Impulse
J = FΔt = Δp
Work
W = Fd cosθ
Kinetic Energy
KE = ½mv²
Grav. PE
PE = mgh
Spring PE
PE = ½kx²
Power
P = W/t = Fv
Centripetal Accel.
ac = v²/r
Centripetal Force
Fc = mv²/r
Universal Gravitation
Fg = Gm₁m₂/r²
Kepler's 3rd Law
T² ∝ r³
Wave Speed
v = fλ
Frequency/Period
f = 1/T
Coulomb's Law
Fe = kq₁q₂/r²
Electric Field
E = F/q
Ohm's Law
V = IR
Electric Power
P = IV = I²R = V²/R
Magnetic Force (charge)
F = qvB sinθ
Magnetic Force (wire)
F = BIL sinθ
Index of Refraction
n = c/v
Snell's Law
n₁sinθ₁ = n₂sinθ₂
Thin Lens/Mirror
1/f = 1/do + 1/di
Magnification
m = −di/do = hi/ho
Photon Energy
E = hf
Photoelectric Effect
KEmax = hf − W0
De Broglie Wavelength
λ = h/p
Mass-Energy Equivalence
E = mc²

Physical Constants

ConstantSymbolValue
Acceleration due to gravityg9.8 m/s²
Universal gravitation constantG6.67×10⁻¹¹ N·m²/kg²
Coulomb's constantk8.99×10⁹ N·m²/C²
Elementary chargee1.6×10⁻¹⁹ C
Planck's constanth6.63×10⁻³⁴ J·s
Speed of light in vacuumc3.0×10⁸ m/s
Speed of sound in air (room temp)v≈343 m/s

Radioactive Decay Types

Decay TypeEmittedMass # ChangeAtomic # ChangePenetration
Alpha (α)⁴₂He nucleus−4−2Stopped by paper (least penetrating, most ionizing)
Beta⁻ (β⁻)⁰₋₁e (electron)0+1Stopped by aluminum
Positron (β⁺)⁰₊₁e0−1Stopped by aluminum
Gamma (γ)⁰₀γ (photon)00Needs lead/concrete (most penetrating, least ionizing)

Half-Life Reference Table

# Half-LivesFraction Left% Remaining% Decayed
01100%0%
11/250%50%
21/425%75%
31/812.5%87.5%
41/166.25%93.75%
51/323.125%96.875%

Series vs Parallel Circuits

PropertySeriesParallel
CurrentSame through every componentDivides among branches
VoltageDivides across componentsSame across every branch
Total resistanceRtotal = R1+R2+...1/Rtotal = 1/R1+1/R2+... (< smallest R)
If one component fails (opens)Entire circuit stopsOther branches keep working

Electromagnetic Spectrum (low → high energy)

RegionRelative WavelengthRelative Frequency/Energy
RadioLongestLowest
MicrowaveLongLow
InfraredMedium-longMedium-low
Visible (ROYGBIV)MediumMedium
UltravioletMedium-shortMedium-high
X-rayShortHigh
GammaShortestHighest

Lens & Mirror Sign Conventions

  1. Converging (convex lens / concave mirror): f is positive.
  2. Diverging (concave lens / convex mirror): f is negative.
  3. Real image: di is positive; can be projected onto a screen.
  4. Virtual image: di is negative; cannot be projected onto a screen.
  5. Upright image: magnification m is positive.
  6. Inverted image: magnification m is negative.
  7. A converging lens/mirror forms a real image only when the object is beyond the focal point (do > f).
  8. A diverging lens/mirror ALWAYS forms a virtual, upright, reduced image, regardless of object distance.

Exam Day Checklist

  • Write down every given value with its units before starting a calculation.
  • Choose the correct kinematic equation based on which variable is missing.
  • Draw a free-body diagram before applying Newton's Second Law.
  • For circular motion, remember the centripetal force is supplied by a real force — identify which one.
  • Check whether a collision is elastic or inelastic before assuming kinetic energy is conserved.
  • In circuits, identify series vs. parallel sections before computing total resistance.
  • Use the right-hand rule carefully for magnetic field and force direction problems.
  • For optics, determine real/virtual and upright/inverted using sign conventions, not guesswork.
  • Track units all the way through a calculation — mismatched units signal an error.
  • Double-check sig figs and rounding only in the final answer, not intermediate steps.
Quick ways to lock in the facts you keep forgetting. Read the big trick, then the small note tells you what it unlocks. Say them out loud — silly is memorable.

Kinematics

Same sign, speeding; opposite, slowing
If velocity and acceleration have the same sign, the object speeds up. Opposite signs mean it's slowing down — direction alone doesn't tell you.
No time? No place? Pick the matching equation
Each kinematic equation is missing exactly one variable — vf=vi+at skips Δx, Δx=vit+½at² skips vf, vf²=vi²+2aΔx skips t. Match what's missing to what you don't have.
Top of the toss, still falling
At the peak of a thrown ball's path, velocity is zero but acceleration is still −g. Zero velocity does NOT mean zero acceleration.
Sideways and downways don't talk
In projectile motion, horizontal and vertical motions are totally independent — horizontal velocity never changes, vertical acceleration is always −g.

Newton's Laws & Forces

Third law pairs never meet on one object
Action-reaction forces act on two DIFFERENT objects, so they can never cancel each other out on the same free-body diagram.
Mass is yours everywhere, weight isn't
Mass (kg) never changes with location. Weight (N) = mg changes wherever gravity changes — that's why you'd weigh less on the Moon but have the same mass.
Friction depends on push-down, not push-across
Friction force = μ×FN depends on how hard surfaces press together (normal force), never on the contact area.
Constant velocity means balanced, not motionless
Zero NET force means constant velocity, which includes rest — but also includes cruising at any constant speed in a straight line.

Momentum & Energy

Momentum always survives, energy sometimes doesn't
Momentum is conserved in EVERY collision. Kinetic energy is only conserved in elastic collisions — inelastic ones lose some to heat/sound.
Perpendicular force does nothing
If a force is at 90° to the displacement, W = Fd cos90° = 0. Carrying a bag level, or moving in a circle — no work is done.
Double the speed, quadruple the energy
KE = ½mv² depends on velocity SQUARED — doubling speed doesn't double kinetic energy, it quadruples it.
Longer time, softer hit
Airbags and padding stretch out the time of impact for the same impulse (Δp), which lowers the force (J=FΔt) and reduces injury.

Circular Motion & Gravity

Centripetal isn't its own force — it's a job title
"Centripetal force" just describes the inward net force — it's actually supplied by tension, gravity, friction, or normal force, never a separate force type.
There's no real push outward
The "flung outward" feeling in a turning car is just your own inertia resisting the turn — there is no real outward (centrifugal) force acting on you.
Double the distance, quarter the pull
Gravity (and Coulomb's law) follow inverse-square laws — doubling distance cuts the force to 1/4, not 1/2.
Farther out, slower orbit
A satellite in a bigger orbit moves slower (v=√(GM/r)) — and its own mass doesn't matter at all.

Waves & Sound

The medium sets the speed limit
Wave speed (v=fλ) depends on the medium only. Change the frequency and wavelength adjusts to compensate — speed doesn't budge.
No stuff, no sound
Sound needs particles to compress and rarefy, so it can't cross a vacuum. Light doesn't need a medium — that's the key difference.
Toward you = squeezed = higher pitch
Doppler effect: a source moving toward you compresses wavefronts (higher perceived pitch); moving away stretches them (lower pitch) — the source's own frequency never changes.

Electricity

Series shares current, parallel shares voltage
In series, the SAME current flows through everything. In parallel, the SAME voltage sits across every branch. Never mix these up.
Parallel resistance always shrinks
Adding a parallel branch gives current another path, so total resistance always drops BELOW the smallest single resistor.
Electric force swings both ways
Unlike gravity (always attractive), Coulomb force can push or pull — like charges repel, opposite charges attract.

Magnetism

Parallel to the field means zero force
F=qvB sinθ — if a charge moves parallel to B (θ=0°), sinθ=0 and the magnetic force vanishes.
No monopoles, ever
You can never isolate just a north or south pole — cut a magnet in half and you get two smaller magnets, each with both poles.
No change, no current
Electromagnetic induction needs a CHANGING magnetic flux — a stationary loop sitting in a constant field induces absolutely nothing.

Optics

Denser bends it in, thinner bends it out
Light bends TOWARD the normal entering a denser (higher-n) medium, and AWAY from the normal entering a thinner one.
Convex mirrors never lie big
A convex (diverging) mirror always makes a virtual, upright, SMALLER image — no exceptions, regardless of object distance.
Real flips it, virtual doesn't
Real images (projectable on a screen) from a single lens or mirror are always inverted. Virtual images are always upright.

Modern Physics

Intensity = more electrons, frequency = stronger electrons
In the photoelectric effect, brighter light ejects MORE electrons; higher-frequency light ejects FASTER (higher-KE) electrons — two different jobs.
Alpha's heavy and slow, gamma's light and fast
Alpha particles are big, stopped by paper, but very ionizing. Gamma rays are pure energy, pass through almost anything, but ionize the least.
Half of what's left, every half-life
Half-life always removes half of whatever remains — it's exponential decay, never a constant number of atoms per unit time.