Unit 3 / Work, Energy, and Power
Unit 3: 35 practice problems
Work from memory first. Draw a diagram, choose a system, and say why your model applies. The 30 selected questions revisit every topic in Work, Energy, and Power; 5 additional problems at the end ask you to explain and apply the models. The worked solutions live on their own page.
This is mixed topic review: short concept checks, calculations, and experiment tasks appear together. Choose a topic below to revisit its lesson, then return to the questions that challenge your model.
Question 1 · AP topic 3.1
1. Topic 3.1 question 4
A 3 kg cart moves at 4 m/s. Find K. What speed would a 12 kg cart need to have the same kinetic energy?
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Question 2 · AP topic 3.2
2. Topic 3.2 question 4
A 5 N horizontal force acts over 4 m while a 1 N opposing force acts throughout. For a 2 kg object starting at rest, find each work, net work, and final speed.
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Question 3 · AP topic 3.3
3. Topic 3.3 question 4
A 300 N/m spring is stretched 0.10 m from relaxed length. Find its energy and the energy after doubling that stretch.
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Question 4 · AP topic 3.4
4. Topic 3.4 question 4
A 2 kg object starts from rest 1.5 m above a floor. Before reaching the floor, 6 J goes into thermal energy. Find its kinetic energy and speed just before landing using g = 10 m/s².
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Question 5 · AP topic 3.5
5. Topic 3.5 question 4
A 10 kg object rises 2 m at constant speed in 4 s. Find work, average power, and instantaneous power if the speed is constant throughout, using g = 10 m/s².
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Question 6 · AP topic 3.1
6. Topic 3.1 question 5
Sketch K versus signed velocity for one mass, marking velocities −2, 0, and +2 m/s. Explain the symmetry and why K isn’t a vector.
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Question 7 · AP topic 3.2
7. Topic 3.2 question 5
A force graph rises linearly from 0 to 6 N over 2 m, then stays at 6 N for another 1 m. Find total work and explain your area decomposition.
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Question 8 · AP topic 3.3
8. Topic 3.3 question 5
Three equal masses m occupy the corners of an equilateral triangle of side r. Write the total gravitational potential energy with zero at infinite separation. Explain the number of terms.
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Question 9 · AP topic 3.4
9. Topic 3.4 question 5
A cart moves between two states with ΔK = +4 J, ΔU = −7 J, and ΔE_thermal = +1 J. Find the external work and explain its sign.
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Question 10 · AP topic 3.5
10. Topic 3.5 question 5
A cart has 20 J of kinetic energy. A constant 5 W input lasts 6 s with 8 J dissipated. Find the final kinetic energy and identify why input power alone doesn’t give the kinetic-energy increase.
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Question 11 · AP topic 3.1
11. Topic 3.1 question 6
A 1 kg cart moves +6 m/s relative to ground. Find its kinetic energy measured from a frame moving +4 m/s, and explain why subtracting ½(1)(4²) from its ground energy would fail.
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Question 12 · AP topic 3.2
12. Topic 3.2 question 6
Compare gravitational work on a book lifted to height h by two different paths. Then compare friction work for two rough paths of different lengths that share endpoints.
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Question 13 · AP topic 3.3
13. Topic 3.3 question 6
An object moves from center distance R to 2R from a planet. Find ΔU symbolically and describe how the general U–r graph represents that change.
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Question 14 · AP topic 3.4
14. Topic 3.4 question 6
Derive maximum compression for a mass m released from rest on a relaxed vertical spring of stiffness k. Compare it with static equilibrium and explain why they differ.
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Question 15 · AP topic 3.5
15. Topic 3.5 question 6
Derive the slope of v² versus time for a mass m driven at constant power P with no losses. Explain how an experiment could use that slope to infer P.
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Question 16 · AP topic 3.1
16. Topic 3.1 question 1
A cart speed doubles while mass stays fixed. What happens to K?
- It doubles
- It quadruples
- It stays the same
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Question 17 · AP topic 3.2
17. Topic 3.2 question 1
A constant force is perpendicular to a particle’s displacement. What work does it do?
- Fd
- Zero
- Negative Fd
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Question 18 · AP topic 3.3
18. Topic 3.3 question 1
You shift the potential-energy zero upward. What changes in a fixed lift?
- The energy values but not ΔU
- The required energy difference
- The gravitational force
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Question 19 · AP topic 3.4
19. Topic 3.4 question 1
For a falling ball–Earth system with negligible air resistance, which account is correct?
- Count gravity work and gravitational-energy loss as separate inputs
- Gravitational energy decreases while kinetic energy increases
- The ball creates energy as it falls
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Question 20 · AP topic 3.5
20. Topic 3.5 question 1
Two useful lifts each raise the same mass through the same height, with equal initial and final speeds and no losses. One takes half the time. Compare average useful output power.
- It is the same
- The faster lift has twice the power
- The faster lift uses half the energy
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Question 21 · AP topic 3.1
21. Topic 3.1 question 2
A cart reverses velocity from +3 to −3 m/s at the same speed. Compare its endpoint kinetic energies.
- The final energy is negative
- The energies are equal
- The final energy is three times larger
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Question 22 · AP topic 3.2
22. Topic 3.2 question 2
What does area under an F_parallel-versus-displacement graph represent?
- Work
- Impulse
- Power
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Question 23 · AP topic 3.3
23. Topic 3.3 question 2
A spring is compressed and stretched by equal magnitudes from relaxed length. Compare U_s.
- Compression has negative energy
- The elastic energies are equal
- Stretching has twice the energy
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Question 24 · AP topic 3.4
24. Topic 3.4 question 2
Mechanical energy decreases because of friction. What follows?
- Total energy is not conserved
- Energy can become thermal energy or sound
- Potential energy must become negative
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Question 25 · AP topic 3.5
25. Topic 3.5 question 2
A constant force acts along a cart’s velocity while the cart speeds up. What happens to instantaneous power?
- It increases
- It remains constant
- It becomes zero
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Question 26 · AP topic 3.1
26. Topic 3.1 question 3
A moving observer measures a different cart speed. How should you find its kinetic energy in that frame?
- Transform velocity, then use ½mv²
- Subtract the observer’s kinetic energy from the ground value
- Keep the ground kinetic energy in every frame
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Question 27 · AP topic 3.2
27. Topic 3.2 question 3
A box is dragged out and back along a rough floor. What can be true?
- Friction work is zero because displacement is zero
- Friction does negative work on both legs
- Friction is conservative
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Question 28 · AP topic 3.3
28. Topic 3.3 question 3
For U_g = −GMm/r with zero at infinity, what happens as r increases?
- U becomes more negative
- U increases toward zero
- U stays constant
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Question 29 · AP topic 3.4
29. Topic 3.4 question 3
A mass compresses a vertical spring while moving downward. Which energy changes must be included?
- Only spring energy
- Both gravitational and spring potential energy
- Only kinetic energy
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Question 30 · AP topic 3.5
30. Topic 3.5 question 3
What does area under a power–time graph give?
- Energy transferred
- Force
- Acceleration
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Explain and apply the model
Try these problems before opening the solutions. State your assumptions, show the diagram or calculation, and explain what makes your answer follow.
Follow a return path
Use g = 10 m/s². A 1.5 kg package is lifted 2.0 m and returned to its initial height. Separately, the package is dragged 3.0 m out and 3.0 m back on a level floor with constant kinetic-friction magnitude 4.0 N. In a third trial an object flies out and back through still air with air resistance throughout. Ignore rotation in these trials.
(a) Calculate gravity work on each leg and over the entire lift-and-return trip. State the change in gravitational potential energy of the package–Earth system after it returns to its initial configuration, and explain the closed-path conservative test.
(b) Calculate friction work on each floor leg and in total. Compare the closed-path effects of kinetic friction and air resistance with gravity, and explain why a configuration potential energy may be assigned to gravity but cannot be assigned to friction or air resistance.
(c) A 5.0 N force separately acts over a 2.0 m straight displacement parallel to it, opposite to it, and perpendicular to it. Calculate the three works and state whether these are scalars or vectors; explain what a negative work value tells you.
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Choose the correct displacement
A constant 10 N horizontal external force pulls an application point on a deformable apparatus 0.60 m right while the center of mass moves 0.20 m right. All other external forces do zero work and give no net horizontal force. In a comparison trial, the same net force moves a structureless, nonrotating 3.0 kg cart 0.60 m on a frictionless level track; it starts from rest.
(a) Calculate the energy transferred through the deformable apparatus’s application point and its change in center-of-mass translational kinetic energy. Quantify the difference and identify possible internal energy changes that can receive it. Explain why using center-of-mass displacement for the force work fails.
(b) Evaluate this claim for the deformable apparatus: “Application-point work must equal only the system’s kinetic-energy change, even when the external force changes configuration.” Give the scientifically valid energy account and name the condition the claim omits.
(c) For the cart trial, first test whether the force application point and center of mass share the same displacement. State the additional object-model assumptions that let net work change only modeled kinetic energy, and then find the final speed. Explain why equal displacement alone would not settle the deformable case.
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Zero work can still turn motion
A structureless 0.50 kg puck travels at 4.0 m/s around a frictionless horizontal circle. Its only unbalanced force is always inward and perpendicular to its instantaneous motion.
(a) Which statement correctly explains the work, velocity direction and kinetic energy during the circular motion?
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Identify a configuration potential
Two approximately spherical masses m₁ = 3.0 kg and m₂ = 4.0 kg interact only through gravity. They change center separation from 0.50 m to 1.0 m. Use G = 6.67 × 10⁻¹¹ N·m²/kg² and choose U = 0 at infinity.
(a) Name the system and conservative interaction that can store potential energy. Calculate both endpoint potential energies and ΔU, and explain which configuration has the larger U.
(b) State whether potential energy is a scalar or vector. If the pair is rotated in space at unchanged separation, predict the change in U and explain why force direction and the sign of U do not supply an energy direction. Then explain whether adding air resistance would erase the system’s gravitational potential energy.
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Choose a mechanical energy inventory
A 0.80 kg ball falls 1.5 m from rest, with g = 10 m/s². Ignore air resistance and rotation. Consider either the structureless ball alone or the ball–Earth system.
(a) Inventory the mechanical energy forms in the single structureless-ball model. Identify the external interaction that changes that inventory, calculate its work and the final kinetic energy, and state what this model omits about a real ball.
(b) For ball plus Earth, calculate ΔU_g and ΔK and find the final speed. Explain why gravity’s work must not be counted as an additional external energy input to this larger system.
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