Unit 3 · 18–23% of the exam Beta

Unit 3 practice test: Work, Energy, and Power

12 exam-style questions from the unit, free, with an explanation for every option. Most have four answer choices, as on the real exam, and a few ask you to type a value; data sets share one table, graph or scene diagram.

Covers: Translational Kinetic Energy; Work; Potential Energy; Conservation of Energy; Power.

Experimental setup

A cart and a spring bumper

A 0.50 kg cart rolls at 2.0 m/s along a level, low-friction track toward a spring bumper (k = 200 N/m) fixed to the end of the track. The cart compresses the spring, stops for an instant, and is pushed back. Take the cart and the spring as the system; friction is negligible.

1. Which bar chart matches the cart-spring system at the instant of greatest compression?

  1. The U_s bar holds the energy; K is zero
  2. K and U_s are equal
  3. The K bar holds the energy; U_s is zero
  4. Both K and U_s are zero, because the cart has stopped
Show the answer

At greatest compression v = 0, so K = 0 and U_s equals the cart's starting kinetic energy, 1.0 J.

  • Correct: The U_s bar holds the energy; K is zero: Right: the cart is momentarily at rest, so its kinetic energy has all become elastic potential energy.
  • K and U_s are equal: At greatest compression the cart is at rest, so K = 0.
  • The K bar holds the energy; U_s is zero: That describes the cart before it touches the spring.
  • Both K and U_s are zero, because the cart has stopped: The energy has not disappeared: it is stored in the spring.

2. At the instant the spring is compressed by half its greatest compression, how do K and U_s compare?

  1. K and U_s are equal
  2. K is three times U_s
  3. U_s is three times K
  4. K is twice U_s
Show the answer

U_s(x_max/2) = ½k(x_max/2)² = ¼ × ½k x_max² = ¼ of the total. K is the remaining ¾, three times U_s.

  • K and U_s are equal: Half the compression is not half the energy: U_s = ½k(Δx)² gives one quarter.
  • Correct: K is three times U_s: Right: U_s ∝ (Δx)², so at half the compression U_s is ¼ of the total and K is ¾.
  • U_s is three times K: At half compression only ¼ of the energy is stored, so K is the larger bar.
  • K is twice U_s: This takes U_s as one third of the total. It is one quarter: (½)².

3. Now take the cart alone as the system. Which describes its energy while it compresses the spring?

  1. Its total energy stays the same, now partly stored as U_s
  2. Its kinetic energy decreases because friction does negative work
  3. Its K decreases, because the spring (external) does negative work on it
  4. Its kinetic energy stays the same until the spring is fully compressed and stops it
Show the answer

Cart alone: only K. The spring is external and pushes opposite to the motion, so it does negative work and ΔK = W_spring < 0.

  • Its total energy stays the same, now partly stored as U_s: U_s belongs to a system that includes the spring. The cart alone has only K.
  • Its kinetic energy decreases because friction does negative work: Friction is negligible here; the spring force is what slows the cart.
  • Correct: Its K decreases, because the spring (external) does negative work on it: Right: the spring's push opposes the motion and takes energy out of the cart system.
  • Its kinetic energy stays the same until the spring is fully compressed and stops it: The spring pushes back the whole time it is compressed, so K falls steadily.

Data table

Running up the stairs

Three students run up the same flight of stairs, rising 4.50 m. Use g = 9.8 m/s².

Stair-climb data
StudentMass (kg)Rise (m)Time (s)
Ana50.04.505.0
Ben75.04.509.0
Cy60.04.506.0

4. Who does the most work against gravity?

  1. Ana, the fastest
  2. Cy
  3. Ben
  4. The three do equal work, since the rise is the same
Show the answer

W = mgh: Ana 2.21 kJ, Ben 3.31 kJ, Cy 2.65 kJ. Ben does the most work.

  • Ana, the fastest: Speed does not change the work against gravity; mgh depends on mass and height.
  • Cy: Cy is heavier than Ana but lighter than Ben.
  • Correct: Ben: Right: the same rise and the largest mass, so the largest mgh.
  • The three do equal work, since the rise is the same: The rise is the same, but mgh also depends on mass.

5. Who has the greatest average power output?

  1. Ben, who does the most work
  2. Ana alone, because she is fastest
  3. The three are equal
  4. Ana and Cy, equally
Show the answer

P = mgh/t: Ana (50.0)(9.8)(4.50)/5.0 = 441 W; Ben 368 W; Cy 441 W.

  • Ben, who does the most work: Ben does the most work but takes the longest, so his rate is lowest.
  • Ana alone, because she is fastest: Cy is heavier and slower in the same proportion: her power matches Ana's.
  • The three are equal: Compute mgh/t: 441 W, 368 W and 441 W.
  • Correct: Ana and Cy, equally: Right: both 441 W; Ben's is about 370 W.

6. In SI base units, the watt in which these powers are measured is

  1. kg·m²/s³
  2. kg·m²/s²
  3. kg·m/s²
  4. kg·m/s
Show the answer

1 W = 1 J/s = 1 kg·m²/s³.

  • Correct: kg·m²/s³: Right: a watt is a joule per second, (kg·m²/s²)/s.
  • kg·m²/s²: That is a joule, the unit of energy, not of power.
  • kg·m/s²: That is a newton, the unit of force.
  • kg·m/s: That is the unit of mass times velocity.

7. A passenger sits still in a train that moves at 25 m/s along a straight track. Which statement about the passenger's kinetic energy is correct?

  1. It is the same measured from the train and from the ground
  2. It is zero in both frames because the passenger is sitting still
  3. It is large in both frames because the train is moving
  4. Zero in the train's frame, large in the ground's
Show the answer

Relative to the train, v = 0 and K = 0. Relative to the ground, v = 25 m/s and K = ½m(25 m/s)². Always say which frame a kinetic energy is measured in.

  • It is the same measured from the train and from the ground: Speed is different in different frames, and K depends on speed.
  • It is zero in both frames because the passenger is sitting still: The passenger is still in the train's frame, but moves at 25 m/s in the ground's frame.
  • It is large in both frames because the train is moving: In the train's frame the passenger has zero speed, so K = 0 there.
  • Correct: Zero in the train's frame, large in the ground's: Right: speed depends on the reference frame, so kinetic energy does too.

8. A worker lowers a 15 kg box onto the floor at constant speed. What sign is the work done by the worker on the box?

  1. Positive, because the worker is making an effort
  2. Zero, because the speed is constant
  3. Positive, because the box moves
  4. Negative, because the push opposes the motion
Show the answer

The worker's force on the box is upward and the displacement is downward (θ = 180°), so the worker does negative work. Gravity does an equal amount of positive work, so the net work is zero and the speed is constant.

  • Positive, because the worker is making an effort: Work in physics is F d cos θ, not effort. Here the force opposes the motion.
  • Zero, because the speed is constant: Constant speed means the net work is zero. The worker's own work is negative, balancing gravity's positive work.
  • Positive, because the box moves: Motion alone does not fix the sign. The angle between the force and the displacement does.
  • Correct: Negative, because the push opposes the motion: Right: force and displacement are opposite, so cos 180° = −1.

9. Which of these forces is nonconservative?

  1. The force from an ideal spring on a block
  2. Kinetic friction from a rough floor on a sliding box
  3. The gravitational force from Earth on a ball
  4. The gravitational force between two distant asteroids
Show the answer

A force is conservative when the work it does depends only on the start and end positions. Kinetic friction does more negative work on a longer path, so it is nonconservative.

  • The force from an ideal spring on a block: An ideal spring is conservative: its work depends only on the start and end stretch.
  • Correct: Kinetic friction from a rough floor on a sliding box: Right: its work depends on the path length, so no potential energy can describe it.
  • The gravitational force from Earth on a ball: Gravity is conservative: its work depends only on the change in height.
  • The gravitational force between two distant asteroids: Gravity is conservative at any distance; its work depends only on the start and end separation.

10. A 0.40 kg cart speeds up from 1.5 m/s to 2.5 m/s. By how much does its kinetic energy change, in joules?

Type a number in J.

Show the answer

ΔK = ½(0.40 kg)[(2.5 m/s)² − (1.5 m/s)²] = 0.20 × (6.25 − 2.25) J = 0.80 J.

  • Answer: 0.80 J

11. A ball on a string moves in a horizontal circle at constant speed. Which is the best explanation of why its speed does not change?

  1. The string's force is perpendicular to the velocity, so it does no work
  2. The string's force is zero, because the ball moves at constant speed
  3. The string does positive work that is canceled by negative work from gravity
  4. The ball has no acceleration, so nothing does work
Show the answer

The velocity is tangent to the circle, and the string's pull has no component along it, so θ = 90° at every instant: W = 0, ΔK = 0, and the speed stays the same.

  • Correct: The string's force is perpendicular to the velocity, so it does no work: Right: a force at 90° to the motion changes the direction but adds no kinetic energy.
  • The string's force is zero, because the ball moves at constant speed: The string pulls the ball inward; without that pull the ball would not turn.
  • The string does positive work that is canceled by negative work from gravity: Gravity is vertical and the circle is horizontal, so gravity does no work either.
  • The ball has no acceleration, so nothing does work: The ball accelerates toward the center. The point is that this force does no work.

12. A 1.0 kg ball is carried from the floor to a 2.0 m high shelf, once straight up and once along a long winding staircase. Compare the work done on the ball by the gravitational force on the two paths.

  1. It is larger on the staircase, which is longer
  2. It is zero on the staircase, because most of the motion is horizontal
  3. It is the same on both paths
  4. It is larger on the straight path, which is steeper
Show the answer

On both paths W_grav = −mgΔy = −(1.0 kg)(9.8 m/s²)(2.0 m) = −20 J. The path does not matter for a conservative force.

  • It is larger on the staircase, which is longer: Path length matters for friction, not for a conservative force like gravity.
  • It is zero on the staircase, because most of the motion is horizontal: Only the horizontal parts do no work; the climbs add up to the same 2.0 m rise.
  • Correct: It is the same on both paths: Right: gravity is conservative, so its work depends only on the change in height.
  • It is larger on the straight path, which is steeper: Steepness changes how fast the height changes, not the total change in height.

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Practice has every question in the unit, with feedback after each one; the notes for every topic are free. Other units: Unit 1 · Unit 2 · Unit 3 · Unit 4 · Unit 5 · Unit 6 · Unit 7 · Unit 8.