Unit Overview

Systems of Particles and Linear Momentum Overview

AP Physics C: Mechanics· 5 min read 📊 14-18% of total AP exam score

1. Unit at a Glance

This unit builds on your knowledge of forces and motion to extend analysis from single particles to multi-particle systems. We first cover how to locate and use the center of mass of a system, then connect force acting over time to momentum change, and finally apply the powerful law of conservation of momentum to a wide range of collision problems. These skills are heavily tested on both multiple-choice and free-response sections of the AP exam.

This unit is split into three core sub-topics:

AP Physics C: Mechanics Center of mass

Learn to calculate center of mass for discrete and continuous systems and analyze system motion relative to the center of mass.

★★⏱ 10 min

AP Physics C: Mechanics Conservation of linear momentum and collisions

Apply conservation of momentum to analyze one- and two-dimensional elastic, inelastic, and perfectly inelastic collisions.

★★★⏱ 15 min

AP Physics C: Mechanics Impulse and momentum

Derive and use the impulse-momentum theorem to relate force, time, and change in an object's momentum.

★★⏱ 12 min

2. Common Pitfalls

Wrong move:

Assuming momentum is conserved in all interactions regardless of external forces.

Why:

Momentum is only conserved for isolated systems with zero net external force.

Correct move:

Always check for net external force on your defined system before applying conservation of momentum.

Wrong move:

Confusing which quantities are conserved for elastic vs perfectly inelastic collisions.

Why:

Kinetic energy is only conserved in elastic collisions, not in inelastic collisions.

Correct move:

Momentum is conserved for all collisions with no net external force; only add the kinetic energy conservation condition for elastic collisions.

3. Quick Reference Cheatsheet

Concept / Formula	Description
$x_{c m} = \frac{1}{M} \sum m_{i} x_{i}$	Center of mass coordinate for discrete system of particles
$ vec{p} = m vec{v}$	Linear momentum of a point mass
$J = \int F d t = Δ p$	Impulse-momentum theorem: impulse equals change in momentum
$m_{1} v_{1 i} + m_{2} v_{2 i} = m_{1} v_{1 f} + m_{2} v_{2 f}$	Conservation of linear momentum for 2-particle collision
$\frac{1}{2} m_{1} v_{1 i}^{2} + \frac{1}{2} m_{2} v_{2 i}^{2} = \frac{1}{2} m_{1} v_{1 f}^{2} + \frac{1}{2} m_{2} v_{2 f}^{2}$	Kinetic energy conservation for elastic collisions
$Δ P_{c m} = M Δ v_{c m} = J_{n e t, e x t}$	Change in center of mass momentum equals net external impulse

What's Next

Start your study of this unit with the first sub-topic on center of mass, the foundational concept for analyzing all multi-particle systems. Once you complete all sub-topics in this unit, you will move on to the next unit on rotation, which builds on the system analysis and conservation skills you learn here.