---
title: "AP Physics C: Mech Unit 3 Review: Work, Energy, and Power"
description: "AP Physics C: Mechanics Unit 3 covers Translational Kinetic Energy, Work, and Potential Energy. Study guides, practice questions, and key terms for every topic."
canonical: "https://fiveable.me/ap-physics-c-mechanics/unit-3"
type: "unit"
subject: "AP Physics C: Mechanics"
unit: "Unit 3 – Work, Energy, and Power"
---

# AP Physics C: Mech Unit 3 Review: Work, Energy, and Power

## Overview

Unit 3 introduces the energy framework that runs through the rest of AP Physics C: Mechanics. You will define kinetic energy with K = (1/2)mv^2, calculate work using the line integral W = integral of F dot dr, connect conservative forces to potential energy functions, apply conservation of mechanical energy with and without nonconservative forces, and find average and instantaneous power.

## AP CED Alignment

This unit hub is organized around AP Course and Exam Description topics, skills, and exam task types when they are available in the source data.
- Topic 3.1: Translational Kinetic Energy
- Topic 3.2: Work
- Topic 3.3: Potential Energy
- Topic 3.4: Conservation of Energy
- Topic 3.5: Power
- Practice 2: Mathematical Routines
- FRQ 1 – Mathematical Routines
- FRQ 4 – Qualitative/Quantitative Translation
- FRQ 2 – Translation Between Representations

## Topics

- [Topic 3.1: Translational Kinetic Energy](/ap-physics-c-mechanics/unit-3/1-translational-kinetic-energy/study-guide/5nq7HC3BH3aW99nk): Define K = (1/2)mv^2, recognize it as a scalar, and explain why its value depends on the observer's reference frame.
- [Topic 3.2: Work](/ap-physics-c-mechanics/unit-3/2-work/study-guide/Y8X1HNf1OSe29MGP): Calculate work using W = Fd cos(theta) for constant forces and W = integral F dot dr for variable forces. Distinguish conservative from nonconservative forces and apply the work-energy theorem.
- [Topic 3.3: Potential Energy](/ap-physics-c-mechanics/unit-3/3-potential-energy/study-guide/I4y3a9MbuG2OgtXM): Derive potential energy functions from conservative forces using delta U = -integral F dot dr, extract force from F_x = -dU/dx, and identify equilibrium from U(x) graphs.
- [Topic 3.4: Conservation of Energy](/ap-physics-c-mechanics/unit-3/4-conservation-of-energy/study-guide/wQp39tHxbOSvmKDT): Apply E = K + U for conservative systems and W_nc = delta E when nonconservative forces act. Choose system boundaries to set up energy equations correctly.
- [Topic 3.5: Power](/ap-physics-c-mechanics/unit-3/5-power/study-guide/aXNJ6Af1NIzpF1IM): Calculate average power with P_avg = W / delta t and instantaneous power with P_inst = dW/dt or P = Fv cos(theta). Convert between watts and other energy-rate units.

## Hardest Topics And Analytics

Snapshot: practice snapshot
This snapshot uses Fiveable practice activity to show where students tend to miss questions and which review moves are worth prioritizing first.
- **61% average MCQ accuracy** (Across 2.9k multiple-choice practice attempts for this unit.)
- **2.9k MCQ attempts** (Practice activity included in this snapshot.)
- **37% average FRQ score** (Across 6 scored free-response attempts for this unit.)
- **Topic 3.3: Potential Energy**: 46% MCQ miss rate across 747 attempts. Review Potential Energy with attention to how the concept appears in AP-style source and evidence questions.
- **Topic 3.5: Power**: 44% MCQ miss rate across 349 attempts. Review Power with attention to how the concept appears in AP-style source and evidence questions.
- **Topic 3.2: Work**: 35% MCQ miss rate across 567 attempts. Review Work with attention to how the concept appears in AP-style source and evidence questions.
- **Topic 3.1: Translational Kinetic Energy**: 32% MCQ miss rate across 689 attempts. Review Translational Kinetic Energy with attention to how the concept appears in AP-style source and evidence questions.

## Review Notes

### Topic 3.1: Translational Kinetic Energy

Translational kinetic energy is the energy an object has due to its translational motion. It is always a scalar and always non-negative. Because kinetic energy depends on v^2, doubling speed quadruples K. The value of K depends on the observer's reference frame, so two inertial observers can legitimately measure different values for the same object.

- **K = (1/2)mv^2**: Kinetic energy in joules; m in kg, v in m/s. Scalar quantity, always greater than or equal to zero.
- **Scalar quantity**: K has magnitude only, no direction. You add kinetic energies algebraically, not as vectors.
- **Frame dependence**: Velocity is measured relative to an observer, so K changes with reference frame. There is no single 'correct' value of K for an object.
- **K vs v graph**: A plot of K versus v is a parabola (quadratic). A plot of K versus m is linear at constant v.

**Checkpoint:** A 2 kg object moves at 3 m/s in one frame and 5 m/s in another. Calculate K in each frame and confirm that both values are physically valid.

Quantity | Depends on mass? | Depends on speed? | Scalar or vector?
--- | --- | --- | ---
Kinetic energy K | Yes, linearly | Yes, quadratically | Scalar
Momentum p | Yes, linearly | Yes, linearly | Vector

### Topic 3.2: Work

Work is the energy transferred into or out of a system by a force acting over a displacement. For a constant force, W = Fd cos(theta), where theta is the angle between force and displacement. For a variable force, W = integral from a to b of F(r) dot dr. The area under a graph of the force component parallel to displacement versus displacement equals the work done. Conservative forces do path-independent work; nonconservative forces like kinetic friction do path-dependent work.

- **W = integral F dot dr**: General definition of work as a line integral. Reduces to Fd cos(theta) when the parallel force component is constant.
- **Conservative force**: A force whose work depends only on initial and final positions, not the path taken. Examples: gravity, spring force. Work around a closed path is zero.
- **Nonconservative force**: A force whose work is path-dependent. Kinetic friction and air resistance are the most common examples; they typically remove mechanical energy from the system.
- **Work-energy theorem**: The net work done on an object equals its change in kinetic energy: W_net = delta K. This holds regardless of whether forces are conservative or not.
- **Sign of work**: Positive work adds energy to the system; negative work removes it. Work is zero when force is perpendicular to displacement.

**Checkpoint:** A spring with k = 200 N/m is compressed 0.10 m from equilibrium. Use the line integral to find the work done by the spring as it returns to equilibrium.

Force type | Path dependent? | Associated potential energy? | Example
--- | --- | --- | ---
Conservative | No | Yes | Gravity, spring
Nonconservative | Yes | No | Kinetic friction, air resistance

### Topic 3.3: Potential Energy

Potential energy is stored energy associated with the configuration of a system whose objects interact through conservative forces. It is a scalar tied to position. The zero reference point is chosen by the analyst to simplify calculation; only changes in potential energy are physically meaningful. The key calculus relationship is delta U = -integral of F_cf dot dr, or equivalently F_x = -dU/dx in one dimension. A U(x) graph encodes both the potential energy and the force at every position.

- **delta U = -integral F_cf dot dr**: Change in potential energy equals the negative of the work done by the conservative force. This is the defining relationship between conservative forces and potential energy.
- **F_x = -dU/dx**: The conservative force in the x-direction equals the negative slope of the U(x) graph. Force points in the direction of decreasing potential energy.
- **Gravitational PE (near surface)**: U_g = mgy, where y is measured from the chosen zero reference. Valid when g is approximately constant.
- **Elastic PE**: U_s = (1/2)k(delta x)^2 for a spring displaced delta x from its natural length.
- **Equilibrium from U(x)**: Equilibrium occurs where dU/dx = 0. A local minimum is stable equilibrium; a local maximum is unstable equilibrium.

**Checkpoint:** Given U(x) = 3x^2 - 12x (in joules, x in meters), find the equilibrium position and determine whether it is stable or unstable.

System | Potential energy formula | Zero reference convention
--- | --- | ---
Object near Earth's surface | U_g = mgy | Chosen height y = 0
Spring-mass system | U_s = (1/2)kx^2 | Natural length x = 0
Two point masses (gravitation) | U = -Gm1m2/r | Zero at r = infinity

### Topic 3.4: Conservation of Energy

Mechanical energy is E = K + U. When only conservative forces act within a chosen system and no work is done on the system from outside, E is constant: K_i + U_i = K_f + U_f. When nonconservative forces act, the work they do equals the change in total mechanical energy: W_nc = delta E = delta K + delta U. Choosing the system boundary carefully determines which forces are internal and which do work across the boundary.

- **E = K + U**: Total mechanical energy is the sum of kinetic and potential energies. Conserved only when all internal forces are conservative and no external work is done.
- **W_nc = delta E**: Work done by nonconservative forces equals the change in total mechanical energy. Friction typically makes delta E negative.
- **System selection**: The choice of system boundary determines which energy transfers appear as work done on the system versus internal energy changes.
- **Energy dissipation**: Nonconservative forces convert mechanical energy into thermal energy or sound. This energy is not lost from the universe but is no longer available as mechanical energy.

**Checkpoint:** A 1 kg block slides down a 2 m ramp inclined at 30 degrees with a coefficient of kinetic friction of 0.2. Use energy methods to find the speed at the bottom.

Scenario | Mechanical energy conserved? | Equation to use
--- | --- | ---
Only conservative forces | Yes | K_i + U_i = K_f + U_f
Friction or drag present | No | K_i + U_i + W_nc = K_f + U_f
External force does work | No | W_ext = delta K + delta U

### Topic 3.5: Power

Power is the rate at which energy is transferred or converted. Average power is P_avg = delta E / delta t or equivalently W / delta t. Instantaneous power is P_inst = dW/dt. When a constant force acts on an object moving at velocity v, instantaneous power simplifies to P_inst = F_parallel * v = Fv cos(theta). The SI unit of power is the watt (1 W = 1 J/s).

- **P_avg = W / delta t**: Average power equals total work divided by elapsed time. Units are watts (W) or joules per second.
- **P_inst = dW/dt**: Instantaneous power is the time derivative of work. Use this when force or velocity varies with time.
- **P_inst = Fv cos(theta)**: For a constant force, instantaneous power equals the dot product of force and velocity. Only the component of force parallel to velocity contributes.
- **Watt**: SI unit of power: 1 W = 1 J/s. Kilowatts (kW) are common in applied problems.

**Checkpoint:** A motor exerts a constant 500 N force on a cart moving at 4 m/s at an angle of 30 degrees between force and velocity. Calculate the instantaneous power delivered.

Formula | When to use | Variables needed
--- | --- | ---
P_avg = W / delta t | Total work over a time interval is known | W, delta t
P_avg = delta E / delta t | Energy change over a time interval is known | delta E, delta t
P_inst = dW/dt | Work is a function of time | W(t) expression
P_inst = Fv cos(theta) | Constant force, known velocity | F, v, theta

## Study Guides

- [3.1 Translational Kinetic Energy](/ap-physics-c-mechanics/unit-3/1-translational-kinetic-energy/study-guide/5nq7HC3BH3aW99nk)
- [3.2 Work](/ap-physics-c-mechanics/unit-3/2-work/study-guide/Y8X1HNf1OSe29MGP)
- [3.3 Potential Energy](/ap-physics-c-mechanics/unit-3/3-potential-energy/study-guide/I4y3a9MbuG2OgtXM)
- [3.4 Conservation of Energy](/ap-physics-c-mechanics/unit-3/4-conservation-of-energy/study-guide/wQp39tHxbOSvmKDT)
- [3.5 Power](/ap-physics-c-mechanics/unit-3/5-power/study-guide/aXNJ6Af1NIzpF1IM)

## Practice Preview

### Multiple-choice practice

- **AP-style practice question**: Practice 2: Mathematical Routines | An object falls through the air subject to a drag force $$F_{drag} = -bv$$, where $$b = 0.50$$ kg/s. The object reaches a constant terminal speed of 20 m/s. What is the rate at which energy is dissipated by the drag force at this speed?
- **AP-style practice question**: Practice 2: Mathematical Routines | A satellite moves in an elliptical orbit around Earth from perigee (closest point) to apogee (farthest point). System I is defined as the satellite only. System II is defined as the satellite and Earth. Which of the following correctly describes the change in total mechanical energy for each system during this motion?
- **AP-style practice question**: Practice 2: Mathematical Routines | A pendulum bob swings downward from its highest point. System A is the bob. System B is the bob and Earth. Which of the following correctly compares the work done by gravity ($$W_{grav}$$) on System A to the change in potential energy ($$\Delta U$$) of System B?
- **AP-style practice question**: Practice 2: Mathematical Routines | A box is lifted vertically at a constant speed by an applied force. System A consists of the box. System B consists of the box and Earth. How does the change in total mechanical energy ($$\Delta E$$) for each system compare to the work done by the applied force ($$W_{app}$$)?
- **AP-style practice question**: Practice 2: Mathematical Routines | A ball is dropped from rest and lands on a vertical spring, compressing it. System I is the ball. System II is the ball, Earth, and spring. Considering the interval from initial contact until the spring reaches maximum compression, which statement is correct?
- **AP-style practice question**: Practice 2: Mathematical Routines | Two blocks connected by a string accelerate across a frictionless table when pulled by a force $$F$$ on the leading block. System I is the leading block. System II is both blocks and the connecting string. How does the change in kinetic energy ($$\Delta K$$) compare to the work done by the force $$F$$ ($$W_F$$)?

### FRQ practice

- **Position-dependent force work and energy**: FRQ 1 – Mathematical Routines | Position-dependent force work and energy
- **Variable force, work-energy theorem, mass comparison**: FRQ 4 – Qualitative/Quantitative Translation | Variable force, work-energy theorem, mass comparison
- **Variable force work and kinetic energy changes**: FRQ 2 – Translation Between Representations | Variable force work and kinetic energy changes

## Key Terms

- **scalar**: A physical quantity described by magnitude only, without direction. Both kinetic energy and work are scalars.

## Common Mistakes

- **Treating work as a vector**: Work is a scalar. The dot product F dot dr produces a signed number, not a vector. Students sometimes try to add work values as vectors, which gives wrong answers.
- **Forgetting the negative sign in F_x = -dU/dx**: The conservative force equals the negative derivative of potential energy. Dropping the negative sign reverses the direction of the force and leads to incorrect equilibrium analysis.
- **Using conservation of mechanical energy when friction is present**: K_i + U_i = K_f + U_f only holds when all forces doing work are conservative. If kinetic friction or drag acts, you must include W_nc = delta E or the answer will be too large.
- **Confusing average power with instantaneous power**: P_avg = W / delta t gives the average rate over an interval. P_inst = Fv cos(theta) gives the rate at one instant. Using the average formula when instantaneous power is asked, or vice versa, produces incorrect results.
- **Setting the wrong zero for potential energy**: The zero reference for U is a free choice, but it must stay consistent throughout a single problem. Switching the reference mid-calculation changes delta U and breaks the energy equation.

## Exam Connections

- **Deriving expressions using calculus**: AP Physics C: Mechanics free-response questions frequently ask you to derive a symbolic expression for work, potential energy, or power using integration or differentiation. You should be comfortable setting up the line integral W = integral F dot dr for a given force function and applying F_x = -dU/dx to extract force from a potential energy expression.
- **Justifying energy conservation claims**: Exam questions often ask you to explain whether mechanical energy is conserved in a given scenario and to justify your answer. You need to identify which forces act, classify them as conservative or nonconservative, and state how that classification determines whether E = K + U is constant or whether W_nc = delta E applies.
- **Interpreting graphs of energy and force**: Questions may present a U(x) graph and ask you to identify equilibrium positions, determine the direction of force at a given point, or sketch the corresponding F(x) graph. You may also be asked to find work as the area under an F vs displacement curve or to read instantaneous power from a slope on an energy versus time graph.

## Final Review Checklist

- **Final Unit 3 review checklist**: Use this list to confirm you can handle every major skill before the exam.
- **Calculate K and explain frame dependence**: Apply K = (1/2)mv^2, confirm the scalar nature of kinetic energy, and explain why two inertial observers can measure different values for the same object.
- **Evaluate work using the line integral**: Set up and solve W = integral F dot dr for variable forces, including spring and gravitational forces. Find work graphically as the area under an F_parallel vs displacement curve.
- **Derive and use potential energy functions**: Use delta U = -integral F_cf dot dr to derive U(x) for gravity and springs. Apply F_x = -dU/dx to recover force from a given potential energy function.
- **Identify equilibrium from U(x) graphs**: Locate points where dU/dx = 0, classify them as stable (local minimum) or unstable (local maximum), and sketch the corresponding force versus position.
- **Apply conservation of energy with and without friction**: Write K_i + U_i = K_f + U_f for conservative systems and include W_nc = delta E when friction or drag is present. Justify your system selection explicitly.
- **Calculate average and instantaneous power**: Use P_avg = W / delta t and P_inst = Fv cos(theta). Derive an instantaneous power expression using calculus when force or velocity is given as a function of time.

## Study Plan

- **Step 1: Kinetic energy and the work-energy theorem (Topics 3.1-3.2)**: Start by confirming you can calculate K = (1/2)mv^2 and explain its scalar nature and frame dependence. Then practice setting up the work integral for both constant and variable forces, including spring forces. Verify the work-energy theorem by checking that W_net = delta K in several examples. Use the Topic 3.1 and 3.2 guides to review the line integral setup.
- **Step 2: Potential energy functions and U(x) graphs (Topic 3.3)**: Practice deriving U(x) from a given conservative force using delta U = -integral F dot dr for gravity and spring systems. Then reverse the process: given a U(x) expression or graph, find F_x = -dU/dx and identify equilibrium points. Sketch force versus position from a U(x) graph. Use the Topic 3.3 guide for worked examples with gravitational and elastic potential energy.
- **Step 3: Conservation of energy problems (Topic 3.4)**: Work through problems that require choosing a system, writing the energy equation, and solving for an unknown speed, height, or compression. Practice both the conservative case (K_i + U_i = K_f + U_f) and the nonconservative case (W_nc = delta E). Use the Topic 3.4 guide and FRQ practice to build fluency with multi-step energy problems.
- **Step 4: Power calculations (Topic 3.5)**: Practice computing average power from total work and time, and instantaneous power using P = Fv cos(theta). Try deriving a power expression as a function of time when force or velocity varies. Use the Topic 3.5 guide to review the calculus-based definition P_inst = dW/dt.
- **Step 5: Full-unit integration and exam practice**: Solve multi-concept problems that combine work, potential energy, conservation of energy, and power in a single scenario. Use the available FRQ practice to rehearse writing complete justifications. Use the AP score calculator to estimate your current scoring range and identify which topics need more attention.

## More Ways To Review

- [Topic study guides](/ap-physics-c-mechanics/unit-3#topics)
- [Practice questions](/ap-physics-c-mechanics/guided-practice?unitSlug=unit-3)
- [FRQ practice](/ap-physics-c-mechanics/frq-practice)
- [Key terms](/ap-physics-c-mechanics/key-terms)

## FAQs

### What topics are covered in AP Physics Mech Unit 3?

AP Physics C: Mechanics Unit 3 covers 5 topics: Translational Kinetic Energy (3.1), Work (3.2), Potential Energy (3.3), Conservation of Energy (3.4), and Power (3.5). The unit builds around the work-energy theorem and conservation principles, which show up repeatedly in later units and on the AP exam. See the full topic breakdown at [AP Physics C: Mechanics Unit 3](/ap-physics-c-mechanics/unit-3).

### How much of the AP Physics Mech exam is Unit 3?

Unit 3: Work, Energy, and Power makes up 15-25% of the AP Physics C: Mechanics exam, making it one of the heavier-weighted units. That percentage covers Translational Kinetic Energy, Work, Potential Energy, Conservation of Energy, and Power. Expect both multiple-choice and free-response questions that test these concepts.

### What's on the AP Physics Mech Unit 3 progress check (MCQ and FRQ)?

The AP Classroom Unit 3 progress check includes both MCQ and FRQ parts drawn from all five Unit 3 topics: Translational Kinetic Energy, Work, Potential Energy, Conservation of Energy, and Power. The MCQ section tests conceptual reasoning and calculation, while the FRQ part asks you to apply the work-energy theorem and conservation of energy to multi-part scenarios. Practice with matched questions at [AP Physics C: Mechanics Unit 3](/ap-physics-c-mechanics/unit-3).

### How do I practice AP Physics Mech Unit 3 FRQs?

Unit 3 FRQs most often come from Conservation of Energy (3.4) and Work (3.2), asking you to set up energy equations, justify sign conventions, and interpret graphs of force vs. displacement. Practice by writing out full solutions that include a clear energy diagram, labeled variables, and a written justification, not just a numerical answer. Find Unit 3 FRQ practice at [AP Physics C: Mechanics Unit 3](/ap-physics-c-mechanics/unit-3).

### Where can I find AP Physics Mech Unit 3 practice questions?

The best place to find AP Physics C: Mechanics Unit 3 practice questions, including multiple-choice and practice test sets, is [AP Physics C: Mechanics Unit 3](/ap-physics-c-mechanics/unit-3). That page organizes MCQ and FRQ practice by topic, covering Translational Kinetic Energy, Work, Potential Energy, Conservation of Energy, and Power so you can target whichever topic needs the most work.

### How should I study AP Physics Mech Unit 3?

Start with Work (3.2) and the work-energy theorem before moving to Potential Energy (3.3) and Conservation of Energy (3.4), since those topics build directly on each other. Sketch energy diagrams for every problem, practice setting up integrals for variable forces, and check your sign conventions carefully. Finish by connecting Power (3.5) to the rate of energy transfer in real scenarios. Use the topic-by-topic resources at [AP Physics C: Mechanics Unit 3](/ap-physics-c-mechanics/unit-3) to track your progress.

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