Review AP Physics C: Mechanics Unit 4 to build a complete understanding of linear momentum, impulse, conservation laws, and collision types. This unit connects Newton's second law to calculus-based momentum analysis and gives you the tools to solve collision and explosion problems systematically.
Use the topic guides, practice questions, and FRQ practice available for this unit to work through momentum and impulse problems with calculus.
Linear momentum is the product of mass and velocity, p = mv, and because velocity is a vector, momentum is too. Unit 4 builds from that definition through impulse, conservation laws, and collision analysis, using calculus at every step.
p = mv means direction matters. When you add momenta in a system, you must account for signs in 1D and use x- and y-components in 2D. A common error is treating momentum as a scalar and losing track of direction after a collision.
The impulse-momentum theorem, J = integral of F_net dt = delta p, is the calculus form of Newton's second law. On a force-time graph, impulse is the area under the curve. On a momentum-time graph, net force is the slope.
In an isolated system, internal forces cancel by Newton's third law, so total momentum is constant. This applies to both collisions and explosions. Choosing the right system boundary is the key skill.
Even when you cannot track every force during a collision, you can compare the total momentum of the system just before and just after the interaction. Because internal forces cancel in pairs, the total is unchanged as long as no net external impulse acts. This makes momentum conservation the primary tool for collision and explosion problems, and it works whether or not kinetic energy is conserved.
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.
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.
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.
Classifies collisions by kinetic energy behavior: elastic (KE conserved), inelastic (KE decreases), and perfectly inelastic (objects stick). Momentum is conserved in all cases.
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Review Change in Momentum and Impulse with attention to how the concept appears in AP-style source and evidence questions.
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.
| Interaction type | Internal forces | Objects after |
|---|---|---|
| Collision | Large, short-duration | Separate or stuck |
| Explosion | Internal forces push apart | Separate, moving in opposite directions |
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.
| 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 |
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.
| 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 |
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.
| 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 |
Try AP-style multiple-choice questions and written prompts after you review the notes.
1. A cart of mass 3.0 kg slides along a horizontal frictionless track in the +x-direction with constant speed 4.0 m/s. The cart collides with a block of mass 1.0 kg that is initially at rest on the track, as shown in Figure 1. During the collision, which lasts from time t = 0 to t = 0.20 s, the cart exerts a force on the block. After the collision, the cart moves in the +x-direction with speed 2.0 m/s and the block moves in the +x-direction.
Figure 1. A 3.0 kg cart moving in the +x-direction at 4.0 m/s approaches a 1.0 kg block initially at rest on a horizontal frictionless track.
Figure 2. Momentum-vector grids for the cart before the collision (given), the block before the collision, and the cart–block system after the collision.
The diagrams in Figure 2 can be used to represent the momentum of the cart and block before the collision and the cart-block system after the collision. The momentum vector diagram for the cart before the collision is shown.
Draw arrows on the grids to represent the momentum vector of the block before the collision and the momentum vector of the cart-block system after the collision.
Arrows should start at the zero-momentum line.
The length of the arrows should be proportional to the relative magnitudes of the vectors.
Represent an arrow of zero length by drawing a dot at zero.
Derive an expression for the speed of the block after the collision. Express your answer in terms of , , , , and physical constants, as appropriate, where = 3.0 kg is the mass of the cart, = 1.0 kg is the mass of the block, = 4.0 m/s is the initial speed of the cart, and = 2.0 m/s is the final speed of the cart. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.
Calculate the numerical value of the speed of the block after the collision.
Describe whether the collision between the cart and block is elastic or inelastic. Justify your answer with a calculation.
During the collision, the force exerted on the block by the cart varies with time according to for , where = 0.20 s is the duration of the collision and is the maximum force exerted on the block.
Derive an expression for in terms of , , , , and physical constants, as appropriate, where is the final speed of the block after the collision. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.
4. Two carts, cart A of mass 2 (see Figure 2) (see Figure 3).0 kg and cart B of mass 3.0 kg, are on a horizontal frictionless track. Cart A moves to the right with an initial speed of 4.0 m/s toward cart B, which is initially at rest, as shown in Figure 1. The carts collide.
Figure 1: Before collision (initial conditions on a frictionless horizontal track)
Figure 2: After elastic collision (Scenario 1)
Figure 3: After perfectly inelastic collision (Scenario 2)
During the collision in Scenario 1, the magnitude of the impulse delivered to cart A by cart B is , and the magnitude of the impulse delivered to cart B by cart A is .
Indicate whether is greater than, less than, or equal to by writing one of the following.
Justify your answer using qualitative reasoning beyond referencing equations.
Derive an expression for the final velocity of cart B after the elastic collision. Express your answer in terms of , , , , and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information. Consider the elastic collision in Scenario 1, where cart A has mass and initial velocity , and cart B has mass and is initially at rest. After the collision, cart A has final velocity .
Indicate whether is greater than, less than, or equal to . Compare the magnitude of the change in momentum of cart A in Scenario 1 (elastic collision) to the magnitude of the change in momentum of cart A in Scenario 2 (perfectly inelastic collision). Let represent the magnitude of the change in momentum of cart A in Scenario 1, and let represent the magnitude of the change in momentum of cart A in Scenario 2.
Briefly justify your answer.
3. A student is given two carts, Cart A of mass mA = 0.50 kg and Cart B of mass mB = 1.0 kg, on a horizontal track. Cart A is initially moving with speed v0 toward Cart B, which is initially at rest.
Students are asked to experimentally determine the coefficient of restitution e for collisions between the two carts using a linear graph. To determine e, the students are permitted to use measurements from only the motion sensor and the meterstick.
Describe an experimental procedure using the described setup to collect data that would allow the students to determine an experimental value of e using a linear graph. Include any steps necessary to reduce experimental uncertainty. In your description, state what quantities would be measured and how the measurements would be made.
Describe how the data collected in part A could be graphed and how that graph would be analyzed to determine the value of e. Include which quantities should be graphed on each axis and how the coefficient of restitution e relates to features of the graph.
Figure 1. Initial setup before collision
Figure 2. After collision
Figure 3. Blank grid for plotting data
vAi (m/s) | vAf (m/s) | vBf (m/s) |
|---|---|---|
0.80 | -0.12 | 0.46 |
1.20 | -0.18 | 0.69 |
1.60 | -0.24 | 0.92 |
2.00 | -0.30 | 1.15 |
2.40 | -0.36 | 1.38 |
The student performs the experiment by giving Cart A various initial speeds vAi while Cart B remains at rest (vBi = 0). For each trial, the student measures the initial speed of Cart A and the final speeds of both carts after the collision. The student's measurements are shown in Table 1.
Indicate two quantities, either measured quantities from Table 1 or additional calculated quantities, that could be graphed to produce a straight line that could be used to determine e.
Vertical axis: Horizontal axis:
On the grid provided in Figure 3, create a graph of the quantities indicated in part C(i).
Use Table 2 to record the measured or calculated quantities that you will plot.
Clearly label the axes, including units as appropriate.
Plot the points you recorded in Table 2.
Draw a best-fit line to the data graphed in part C(ii).
Using the best-fit line that you drew in part C(iii), calculate an experimental value for the coefficient of restitution e. Show your work clearly.
| Term | Definition |
|---|---|
| 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. |
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.
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.
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.
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.
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.
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.
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.
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.
Open the individual guides for Unit 4 when you want a closer review of one topic.
browse guidesUse AP-style practice after you review the notes so you can check what you understand.
start practicePractice free-response reasoning and compare your answer with scoring guidance.
practice FRQsUse unit cheatsheets for a quick visual review after you work through the notes.
open cheatsheetsEstimate your broader AP score goal after you review the course and exam format.
open calculatorAP 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.
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.
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.
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.
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. 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.
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.