Unit 4: Linear Momentum

What Unit 4 Covers

Unit 4 is the momentum unit: it introduces linear momentum $p = mv$, impulse as the way momentum changes, and conservation of momentum as the tool for analyzing collisions and explosions. On the AP Physics 1 exam it carries 10 to 15 percent of the multiple-choice section, the same weight as kinematics.

The unit has four topics. Topic 4.1 defines momentum, 4.2 connects force and time to momentum change through impulse, 4.3 establishes when a system's total momentum stays constant, and 4.4 sorts collisions into elastic and inelastic types. Everything builds toward one skill: picking a system, checking whether external forces matter, and writing a before-and-after momentum equation.

If Unit 3 taught you to track energy, Unit 4 gives you a second conservation law. Plenty of exam questions require both at once, so treat the two units as a pair.

Topic 4.1: Linear Momentum

Momentum is mass times velocity: $p = mv$, measured in kg·m/s. Unlike kinetic energy, momentum is a vector, so direction matters and you must assign a positive direction before plugging in numbers. A 2 kg cart moving left at 3 m/s has momentum $-6$ kg·m/s if right is positive.

For a system of objects, total momentum is the vector sum of the individual momenta. This is the idea the rest of the unit leans on: exam questions often ask about the momentum of a two-cart system rather than a single object, so get comfortable adding signed momenta.

A useful contrast to internalize: two objects can have equal kinetic energy but different momenta, and the reverse. A question that gives you kinetic energy and asks about momentum is testing whether you can move between $K = \frac{1}{2}mv^2$ and $p = mv$ by combining the two equations.

Topic 4.2: Change in Momentum and Impulse

Impulse is the product of average force and the time interval over which it acts: $J = F_{avg}\Delta t = \Delta p$. This impulse-momentum theorem is the unit's main link back to Newton's second law: a net force applied over time changes momentum.

The exam's favorite representation here is the force-time graph. Impulse equals the area under the curve, so a triangular force pulse peaking at 100 N over 0.2 s delivers $\frac{1}{2}(100)(0.2) = 10$ N·s. Expect at least one graph-reading question, and expect piecewise or curved graphs where you estimate area by splitting shapes or counting grid boxes.

The same equation explains cushioning: airbags and crumple zones stretch out $\Delta t$ for a fixed $\Delta p$, so $F_{avg}$ drops. The full derivation and more graph practice are in the impulse-momentum theorem guide.

Topic 4.3: Conservation of Linear Momentum

The total momentum of a system is constant when the net external force on the system is zero. That single sentence is Topic 4.3. Internal forces, like the contact forces two colliding carts exert on each other, come in Newton's third law pairs and cancel, so they can never change the system's total momentum.

The working method: choose your system so the interaction is internal, set a positive direction, then write $m_1v_1 + m_2v_2 = m_1v_1' + m_2v_2'$ with a sign attached to every velocity. A dropped negative sign is the error to check for on every collision problem.

Conservation also covers explosions and recoil. A system that starts at rest must end with zero total momentum, so the pieces move in opposite directions with speeds in inverse proportion to their masses. Step-by-step setups for collisions, explosions, and recoil are in the conservation of momentum guide.

Topic 4.4: Elastic and Inelastic Collisions

Every collision conserves momentum when external forces are negligible, so the classification in Topic 4.4 is entirely about kinetic energy:

• Elastic: total kinetic energy is the same before and after. Carts with magnetic bumpers approximate this.
• Inelastic: some kinetic energy converts to other forms. Most real collisions fall here.
• Perfectly inelastic: the objects stick together and share one final velocity, which gives the maximum kinetic energy loss consistent with momentum conservation.

The tested skill is computing $K$ before and after and comparing, or reasoning qualitatively about which case applies. If a problem says objects couple, latch, or embed, it is perfectly inelastic, and one unknown disappears because both objects share the same final velocity.

Run scenarios through the momentum collision calculator to see how final speeds and energy loss change with mass ratio and collision type.

How Unit 4 Is Tested

Unit 4 supplies 10 to 15 percent of Section I, which has 42 multiple-choice questions in 85 minutes. Momentum also shows up in Section II, where the four free-response formats are Mathematical Routines, Translation Between Representations, Experimental Design and Analysis, and Qualitative/Quantitative Translation. Collisions fit Mathematical Routines naturally, and force-time graphs are classic Translation material. A calculator is allowed on both sections.

Common question patterns:

• Read impulse off a force-time graph, then find a final velocity.
• Analyze a two-object collision with a before-and-after table, then classify it by comparing kinetic energy.
• Design an experiment to test whether momentum is conserved on a track.
• Explain qualitatively why an airbag reduces the average force on a passenger.

Multi-step problems often chain a collision into an energy calculation, like a rise up a ramp or a spring compression, which pulls in conservation of energy as well.

Study Plan and Practice

Work the unit in order: definition, impulse, conservation, classification. For each topic, solve a few problems by hand, then verify your answers.

1. Read the conservation of momentum guide and the impulse-momentum theorem guide for the two core skills.
2. Practice graph problems until finding the area under a force-time curve is automatic.
3. Use the momentum collision calculator to check hand solutions and build intuition for elastic vs perfectly inelastic outcomes.
4. Keep the AP Physics 1 formula sheet nearby so you know exactly which equations are printed: $p = mv$ and $J = F_{avg}\Delta t = \Delta p$ both are.
5. Review Unit 3 energy methods, since collision problems often end with an energy step.

Momentum ideas return in Unit 6 as angular momentum, so mastering signs and system choice here pays off twice.