---
title: "AP Physics C: Mechanics Unit 4 Review: Linear Momentum"
description: "AP Physics C: Mechanics Unit 4 covers Linear Momentum and Change in Momentum and Impulse. Study guides, practice questions, and key terms for every topic."
canonical: "https://fiveable.me/ap-physics-c-mechanics/unit-4"
type: "unit"
subject: "AP Physics C: Mechanics"
unit: "Unit 4 – Linear Momentum"
---

# AP Physics C: Mechanics Unit 4 Review: Linear Momentum

## Overview

Unit 4 covers linear momentum as a vector quantity, the impulse-momentum theorem derived from calculus, conservation of momentum in isolated systems, and the distinction between elastic and inelastic collisions. The unit carries 10-20% of the AP Physics C: Mechanics exam and requires both conceptual reasoning and calculus-based problem solving.

## 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 4.1: Linear Momentum
- Topic 4.2: Change in Momentum and Impulse
- Topic 4.3: Conservation of Linear Momentum
- Topic 4.4: Elastic and Inelastic Collisions
- Practice 3: Scientific Questioning and Argumentation
- FRQ 1 – Mathematical Routines
- FRQ 4 – Qualitative/Quantitative Translation
- FRQ 3 – Experimental Design

## Topics

- [Topic 4.1: Linear Momentum](/ap-physics-c-mechanics/unit-4/1-linear-momentum/study-guide/vUy5cuBKbgKou3xJ): Defines momentum as p = mv, establishes it as a vector quantity, and introduces the collision and explosion models for analyzing interactions by comparing initial and final system states.
- [Topic 4.2: Change in Momentum and Impulse](/ap-physics-c-mechanics/unit-4/2-change-in-momentum-and-impulse/study-guide/dj7haZ7RqOOTuCat): Derives impulse as the integral of net force over time and connects it to change in momentum through the impulse-momentum theorem. Covers graphical interpretation of force-time and momentum-time graphs.
- [Topic 4.3: Conservation of Linear Momentum](/ap-physics-c-mechanics/unit-4/3-conservation-of-linear-momentum/study-guide/qWLd3tCmiQRSZKv0): Establishes that total momentum is constant in isolated systems, introduces center-of-mass velocity, and applies conservation component-wise to 1D and 2D collisions and explosions.
- [Topic 4.4: Elastic and Inelastic Collisions](/ap-physics-c-mechanics/unit-4/4-elastic-and-inelastic-collisions/study-guide/JvjEzhTLprTtp64U): Classifies collisions by kinetic energy behavior: elastic (KE conserved), inelastic (KE decreases), and perfectly inelastic (objects stick). Momentum is conserved in all cases.

## 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.
- **62% average MCQ accuracy** (Across 1.9k multiple-choice practice attempts for this unit.)
- **1.9k MCQ attempts** (Practice activity included in this snapshot.)
- **75% average FRQ score** (Across 2 scored free-response attempts for this unit.)
- **Topic 4.2: Change in Momentum and Impulse**: 35% MCQ miss rate across 547 attempts. Review Change in Momentum and Impulse with attention to how the concept appears in AP-style source and evidence questions.

## Review Notes

### Topic 4.1: Linear Momentum

Linear momentum is defined as p = mv. Because velocity is a vector, momentum has both magnitude and direction. For a system of objects, total momentum is the vector sum of individual momenta. Collisions and explosions are modeled by comparing the system's momentum just before and just after the interaction, treating each object as a point mass.

- **p = mv**: Linear momentum equals mass times velocity; direction matches the direction of velocity.
- **Vector addition**: System momentum is the sum of all individual momenta, requiring component-wise addition in 2D problems.
- **Collision model**: Internal forces during the interaction are much larger than external forces, so only initial and final states need to be analyzed.
- **Explosion model**: Internal forces push system objects apart; total momentum before and after is still compared using the same framework.
- **Object model**: Each object is treated as a point mass during a collision, ignoring size and internal structure.

**Checkpoint:** If a 2 kg object moves at 3 m/s east and a 1 kg object moves at 6 m/s west, what is the total momentum of the system?

Interaction type | Internal forces | Objects after
--- | --- | ---
Collision | Large, short-duration | Separate or stuck
Explosion | Internal forces push apart | Separate, moving in opposite directions

### Topic 4.2: Change in Momentum and Impulse

The net external force on a system equals the rate of change of its momentum: F_net = dp/dt. Impulse J is the integral of net force over a time interval, J = integral from t1 to t2 of F_net(t) dt, and equals the change in momentum delta p. Impulse is a vector in the direction of the net force. Graphically, impulse is the area under a force-time curve, and net force is the slope of a momentum-time graph.

- **F_net = dp/dt**: Newton's second law in momentum form; for constant mass this reduces to F_net = ma.
- **J = integral of F_net dt = delta p**: The impulse-momentum theorem: impulse equals change in momentum, linking force, time, and velocity change.
- **Area under F vs. t graph**: The area under the curve of a force-time graph gives the impulse delivered during that interval.
- **Slope of p vs. t graph**: The slope of a momentum-time graph at any point equals the net external force at that instant.
- **Variable mass case**: When mass changes with time, F_net = dp/dt = (dm/dt)v, which applies to rocket propulsion problems.

**Checkpoint:** A force varies with time as shown on a graph. How would you find the impulse delivered between t = 0 and t = 4 s without a constant force?

Graph type | Slope represents | Area represents
--- | --- | ---
Force vs. time | Rate of change of force | Impulse (delta p)
Momentum vs. time | Net external force | Not directly used

### Topic 4.3: Conservation of Linear Momentum

When the net external force on a system is zero, total momentum is conserved: the sum of initial momenta equals the sum of final momenta. Internal forces between objects cancel by Newton's third law, so they do not change total system momentum. The center-of-mass velocity is v_cm = (sum of m_i * v_i) / (sum of m_i), and it remains constant in the absence of a net external force. Conservation applies component-wise in 2D problems.

- **Isolated system**: A system with no net external force, so total momentum is constant throughout any internal interaction.
- **v_cm = sum(m_i * v_i) / sum(m_i)**: Center-of-mass velocity equals total momentum divided by total mass; constant when no net external force acts.
- **Newton's third law cancellation**: Action-reaction force pairs are internal to the system and cancel, leaving only external forces to change total momentum.
- **Component-wise conservation**: In 2D collisions, momentum is conserved separately in the x- and y-directions, giving two independent equations.
- **System selection**: Choosing which objects to include in the system determines which forces are internal and which are external.

**Checkpoint:** A 3 kg cart moving at 4 m/s east collides with a stationary 1 kg cart. If they stick together, what is the final velocity of the combined system?

Condition | Is momentum conserved? | Reason
--- | --- | ---
No net external force | Yes | Internal forces cancel by Newton's third law
Net external force present | No | External impulse changes total momentum
Explosion (internal forces only) | Yes | No net external force on the system

### Topic 4.4: Elastic and Inelastic Collisions

Momentum is conserved in all collisions when the system is isolated. The collision type is determined by what happens to kinetic energy. In an elastic collision, total kinetic energy is the same before and after. In an inelastic collision, total kinetic energy decreases because nonconservative forces convert some kinetic energy into heat, sound, or deformation. In a perfectly inelastic collision, the objects stick together and share a common final velocity.

- **Elastic collision**: Both momentum and total kinetic energy are conserved; individual object kinetic energies may change.
- **Inelastic collision**: Momentum is conserved but total kinetic energy decreases; energy is lost to nonconservative processes.
- **Perfectly inelastic collision**: Objects stick together after the collision and move with the same final velocity; maximum kinetic energy is lost.
- **Kinetic energy check**: Compare (1/2)mv^2 totals before and after; if equal, elastic; if less after, inelastic.
- **Two equations for elastic 1D**: Use conservation of momentum and conservation of kinetic energy together to solve for two unknown final velocities.

**Checkpoint:** Two objects collide and stick together. Is kinetic energy conserved? How do you find the final velocity?

Collision type | Momentum conserved? | KE conserved? | Objects after
--- | --- | --- | ---
Elastic | Yes | Yes | Separate
Inelastic | Yes | No (decreases) | Separate
Perfectly inelastic | Yes | No (maximum loss) | Stuck together

## Study Guides

- [4.1 Linear Momentum](/ap-physics-c-mechanics/unit-4/1-linear-momentum/study-guide/vUy5cuBKbgKou3xJ)
- [4.2 Change in Momentum and Impulse](/ap-physics-c-mechanics/unit-4/2-change-in-momentum-and-impulse/study-guide/dj7haZ7RqOOTuCat)
- [4.3 Conservation of Linear Momentum](/ap-physics-c-mechanics/unit-4/3-conservation-of-linear-momentum/study-guide/qWLd3tCmiQRSZKv0)
- [4.4 Elastic and Inelastic Collisions](/ap-physics-c-mechanics/unit-4/4-elastic-and-inelastic-collisions/study-guide/JvjEzhTLprTtp64U)

## Practice Preview

### Multiple-choice practice

- **AP-style practice question**: Practice 3: Scientific Questioning and Argumentation | A 0.50 kg ball strikes a wall horizontally at 10 m/s and rebounds in the opposite direction at 8.0 m/s. The contact lasts 0.10 s. A student claims the average force exerted by the wall has a magnitude of 90 N. Which of the following correctly evaluates this claim with appropriate justification?
- **AP-style practice question**: Practice 3: Scientific Questioning and Argumentation | Object A of mass $$m$$ moves at speed $$v$$ and collides with identical Object B at rest. They scatter at $$45^\circ$$ angles to the original direction with equal speeds. A student claims the collision is elastic. Which reasoning supports this?
- **AP-style practice question**: Practice 3: Scientific Questioning and Argumentation | A student analyzes a collision between two carts in the center-of-mass reference frame. The carts approach, collide, and stop dead. The student claims this is perfectly inelastic. Which reasoning supports this?
- **AP-style practice question**: Practice 3: Scientific Questioning and Argumentation | Object 1 ($$m$$, $$v$$) and Object 2 ($$3m$$, $$-v$$) collide. Post-collision, Object 1 moves at $$-2v$$ and Object 2 stops. A student claims this collision is elastic. Which reasoning supports this?
- **AP-style practice question**: Practice 3: Scientific Questioning and Argumentation | A rubber ball bounces off a wall. A clay ball of equal mass hits the wall at the same speed and sticks. A student claims the rubber ball collision is more elastic. Which justification is correct?
- **AP-style practice question**: Practice 3: Scientific Questioning and Argumentation | A force-compression graph for a collision bumper shows a hysteresis loop where the force is higher during compression than expansion. A student claims the collision is inelastic. Which reasoning uses the graph to support this?

### FRQ practice

- **Collision elasticity and impulse-momentum analysis**: FRQ 1 – Mathematical Routines | Collision elasticity and impulse-momentum analysis
- **Elastic collision between two carts momentum conservation**: FRQ 4 – Qualitative/Quantitative Translation | Elastic collision between two carts momentum conservation
- **Collision coefficient of restitution determination**: FRQ 3 – Experimental Design | Collision coefficient of restitution determination

## Key Terms

- **collision**: A model for an interaction between objects in which the internal forces are much larger than any net external force, so only initial and final states need to be analyzed.
- **explosion**: A model for an interaction in which forces internal to a system push objects within that system apart; total system momentum is conserved.
- **impulse-momentum theorem**: States that the impulse delivered to an object equals its change in momentum: J = integral of F_net dt = delta p.
- **perfectly inelastic collision**: A collision in which the objects stick together and move with the same final velocity; kinetic energy loss is at its maximum for that momentum exchange.
- **area under the curve**: On a force-time graph, the area under the curve equals the impulse delivered to the object during that time interval.
- **object model**: A simplifying model that treats each object as a point mass, ignoring size and internal structure, which is valid for analyzing collisions.

## Common Mistakes

- **Ignoring the vector nature of momentum**: Momentum has direction. In 1D problems, assign positive and negative signs based on direction before calculating. In 2D, split into x- and y-components and conserve each separately.
- **Confusing impulse with force**: Impulse is the integral of force over time, not force alone. A large force over a short time and a small force over a long time can deliver the same impulse. Always multiply or integrate with the time interval.
- **Assuming kinetic energy is conserved in all collisions**: Kinetic energy is only conserved in elastic collisions. In any inelastic collision, including perfectly inelastic ones, total kinetic energy decreases. Momentum is always conserved in an isolated system regardless of collision type.
- **Applying conservation of momentum when a net external force acts**: Conservation of momentum requires that the net external impulse on the system is zero. If gravity, friction, or another external force acts during the interaction and is not negligible, total momentum changes.
- **Using the wrong final velocity in a perfectly inelastic collision**: When objects stick together, they share one final velocity. Set up m1*v1 + m2*v2 = (m1 + m2)*v_f and solve for v_f. Do not use separate final velocities for each object.

## Exam Connections

- **Calculus-based impulse calculations**: AP Physics C: Mechanics problems frequently give a force as a function of time and ask you to integrate to find impulse, or present a force-time graph and ask for the area. Be ready to set up and evaluate definite integrals and to read graphical data accurately.
- **Multi-step collision problems combining momentum and energy**: Free-response problems often require you to apply conservation of momentum to find a post-collision velocity, then use that result in a separate energy or kinematics calculation. Recognizing which conservation law applies at each stage is the central skill.
- **Justifying whether momentum is conserved**: Exam questions ask you to explain in words why momentum is or is not conserved in a given scenario. A complete answer names the system, identifies whether a net external force acts, and references Newton's third law for internal force cancellation.

## Final Review Checklist

- **Define and calculate momentum as a vector**: Use p = mv with correct direction. For systems, sum momenta as vectors, using components in 2D problems.
- **Apply the impulse-momentum theorem with calculus**: Calculate impulse as the integral of F_net over time, or read it as the area under a force-time graph. Confirm J = delta p.
- **Read force-time and momentum-time graphs correctly**: Area under a force-time curve gives impulse. Slope of a momentum-time curve gives net force. Practice both directions.
- **Apply conservation of momentum to collisions and explosions**: Identify the system, confirm no net external force, then set total initial momentum equal to total final momentum. Use x- and y-components for 2D.
- **Classify collisions and solve for unknowns**: Check whether kinetic energy is conserved to identify elastic vs. inelastic. For perfectly inelastic, use one shared final velocity. For elastic 1D, use two equations.
- **Calculate center-of-mass velocity**: Use v_cm = sum(m_i * v_i) / sum(m_i). Recognize that v_cm is constant when no net external force acts on the system.

## Study Plan

- **Start with momentum as a vector (Topic 4.1)**: Read the Topic 4.1 guide and practice writing p = mv with correct direction for single objects and systems. Sketch momentum vectors for collision and explosion scenarios before and after the interaction.
- **Work through impulse and graphical interpretation (Topic 4.2)**: Use the Topic 4.2 guide to practice calculating impulse as an integral and as the area under a force-time graph. Then practice finding net force as the slope of a momentum-time graph. Try at least one non-constant force problem.
- **Practice conservation of momentum in 1D and 2D (Topic 4.3)**: Use the Topic 4.3 guide to set up conservation equations for collisions and explosions. Practice identifying the system boundary, then solve 1D problems before moving to 2D component problems.
- **Classify and solve collision problems (Topic 4.4)**: Use the Topic 4.4 guide to practice identifying elastic, inelastic, and perfectly inelastic collisions. For each type, write the correct set of equations and solve for unknown velocities. Check kinetic energy before and after to confirm your classification.
- **Review with practice questions and FRQ practice**: Work through the 25+ available practice questions and FRQ practice problems for this unit. Focus on multi-part problems that combine impulse, conservation of momentum, and collision classification in a single scenario. Use the AP score calculator to estimate your overall exam performance.

## More Ways To Review

- [Topic study guides](/ap-physics-c-mechanics/unit-4#topics)
- [Practice questions](/ap-physics-c-mechanics/guided-practice?unitSlug=unit-4)
- [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 4?

AP Physics C: Mechanics Unit 4 covers four topics: Linear Momentum (4.1), Change in Momentum and Impulse (4.2), Conservation of Linear Momentum (4.3), and Elastic and Inelastic Collisions (4.4). The unit builds from defining momentum as mass times velocity up through analyzing real collisions using conservation laws. Here's a quick breakdown:
- **4.1 Linear Momentum**. defining p = mv and its vector nature
- **4.2 Change in Momentum and Impulse**. the impulse-momentum theorem and force-time relationships
- **4.3 Conservation of Linear Momentum**. applying conservation in isolated systems
- **4.4 Elastic and Inelastic Collisions**. distinguishing collision types and solving for unknowns See all four topics at [AP Physics C: Mechanics Unit 4](/ap-physics-c-mechanics/unit-4).

### How much of the AP Physics Mech exam is Unit 4?

Unit 4: Linear Momentum makes up 10-20% of the AP Physics C: Mechanics exam, making it one of the more heavily weighted units. It covers momentum, impulse, conservation of linear momentum, and elastic and inelastic collisions. Expect multiple-choice questions and at least one free-response problem drawing from these concepts. Because the exam weight is significant, it's worth spending real time on conservation of momentum and collision analysis, since those show up most often in both MCQ and FRQ sections.

### What's on the AP Physics Mech Unit 4 progress check (MCQ and FRQ)?

The AP Physics C: Mechanics Unit 4 progress check in AP Classroom includes both MCQ and FRQ parts drawn from all four unit topics: Linear Momentum, Change in Momentum and Impulse, Conservation of Linear Momentum, and Elastic and Inelastic Collisions. The MCQ section tests conceptual understanding and calculation, while the FRQ part asks you to set up and solve multi-step problems involving impulse-momentum and collision scenarios. For the progress check, focus on applying the impulse-momentum theorem correctly and distinguishing elastic from inelastic collisions. You can find matched practice problems at [AP Physics C: Mechanics Unit 4](/ap-physics-c-mechanics/unit-4).

### How do I practice AP Physics Mech Unit 4 FRQs?

Unit 4 FRQs in AP Physics C: Mechanics most often come from Conservation of Linear Momentum and Elastic and Inelastic Collisions, so those two topics should anchor your free-response practice. A typical FRQ will ask you to define a system, justify why momentum is conserved, set up conservation equations, and solve for an unknown velocity or energy loss. To practice effectively:
- Write out your system definition and conservation statement before calculating
- Show all vector directions explicitly, since momentum is a vector quantity
- Practice both perfectly inelastic and elastic collision setups, as each has a different equation structure
- Check your work by verifying units and the reasonableness of your answer Find FRQ-style practice problems at [AP Physics C: Mechanics Unit 4](/ap-physics-c-mechanics/unit-4).

### Where can I find AP Physics Mech Unit 4 practice questions?

The best place to find AP Physics C: Mechanics Unit 4 practice questions, including multiple-choice and practice test problems, is [AP Physics C: Mechanics Unit 4](/ap-physics-c-mechanics/unit-4). That page has resources covering all four topics: Linear Momentum, Change in Momentum and Impulse, Conservation of Linear Momentum, and Elastic and Inelastic Collisions. For MCQ practice, look for questions that ask you to calculate impulse, compare momentum before and after a collision, or identify whether kinetic energy is conserved. These are the most common question formats for this unit on the actual exam.

### How should I study AP Physics Mech Unit 4?

Start with the impulse-momentum theorem in Topic 4.2 before moving to conservation laws, since understanding how force and time change momentum makes Topic 4.3 much more intuitive. Unit 4 rewards students who practice setting up problems systematically rather than jumping straight to equations. A solid study plan for this unit:
1. **Build the foundation first.** Make sure you can define momentum as a vector and apply p = mv in multiple dimensions before tackling collisions.
2. **Master the impulse-momentum theorem.** Practice interpreting force-time graphs and calculating impulse from area under the curve.
3. **Drill conservation of momentum.** Write out system definitions and justify conservation conditions every time, not just when the problem asks you to.
4. **Separate collision types clearly.** Know that elastic collisions conserve both momentum and kinetic energy, while inelastic collisions conserve only momentum.
5. **Do timed FRQ practice.** Since this unit is 10-20% of the exam, free-response problems here are high value. All four topics are organized at [AP Physics C: Mechanics Unit 4](/ap-physics-c-mechanics/unit-4).

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Here's a quick breakdown:\n- **4.1 Linear Momentum**. defining p = mv and its vector nature\n- **4.2 Change in Momentum and Impulse**. the impulse-momentum theorem and force-time relationships\n- **4.3 Conservation of Linear Momentum**. applying conservation in isolated systems\n- **4.4 Elastic and Inelastic Collisions**. distinguishing collision types and solving for unknowns See all four topics at <a href=\"/ap-physics-c-mechanics/unit-4\">AP Physics C: Mechanics Unit 4</a>."}},{"@type":"Question","@id":"https://fiveable.me/ap-physics-c-mechanics/unit-4#how-much-of-the-ap-physics-mech-exam-is-unit-4","name":"How much of the AP Physics Mech exam is Unit 4?","acceptedAnswer":{"@type":"Answer","text":"Unit 4: Linear Momentum makes up 10-20% of the AP Physics C: Mechanics exam, making it one of the more heavily weighted units. It covers momentum, impulse, conservation of linear momentum, and elastic and inelastic collisions. 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That page has resources covering all four topics: Linear Momentum, Change in Momentum and Impulse, Conservation of Linear Momentum, and Elastic and Inelastic Collisions. For MCQ practice, look for questions that ask you to calculate impulse, compare momentum before and after a collision, or identify whether kinetic energy is conserved. These are the most common question formats for this unit on the actual exam."}},{"@type":"Question","@id":"https://fiveable.me/ap-physics-c-mechanics/unit-4#how-should-i-study-ap-physics-mech-unit-4","name":"How should I study AP Physics Mech Unit 4?","acceptedAnswer":{"@type":"Answer","text":"Start with the impulse-momentum theorem in Topic 4.2 before moving to conservation laws, since understanding how force and time change momentum makes Topic 4.3 much more intuitive. Unit 4 rewards students who practice setting up problems systematically rather than jumping straight to equations. 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