AP Physics C: Mechanics · Unit 4: Linear Momentum · Lesson 4.1

Deep Dive: Linear Momentum

🔬 Deep Dive
This is your textbook for this topic. Take your time. Read it more than once.
4.1.A.14.1.A.2Math

Defining Linear Momentum

Linear momentum is a new vector quantity, defined simply as the product of an object's mass and velocity:

p⃗ = mv⃗
🔑Because velocity is a vector, momentum inherits its direction exactly — momentum always points the same way the object is actually moving. There's no scenario where momentum points anywhere else.
Moving rightp⃗Moving up-leftp⃗Momentum's direction always matches velocity's direction — nothing else.

Explore this directly below.

Adjust mass, speed, and direction. The momentum vector always points exactly where the object is heading — only its length depends on mass and speed.

mass (kg)6
speed (m/s)8
direction30°
p⃗ direction = v⃗ direction
|p⃗| = (6)(8) = 48.0 kg·m/s, at 30°
4.1.A.34.1.A.3.i4.1.A.3.iiConcept

Collisions as a Model

Momentum is the natural tool for analyzing collisions — a model for any interaction where the forces exerted between the objects involved are much larger than any net external force acting on them during the interaction itself.

🔑This is exactly why the object model works so well for collisions: since the interaction is so brief and so intense compared to everything else, you only need to track the initial and final states — what happens during the split-second of contact itself doesn't need to be modeled in detail.

Two carts colliding on a track — exactly like this unit's header art — is the classic example: whatever normal forces or friction the track exerts externally are negligible compared to the contact force between the carts themselves during the collision.

4.1.A.3.iiiConcept

Explosions as a Model

An explosion is the mirror image of a collision: a model for an interaction in which forces internal to a system push objects within that system apart, rather than bringing them together.

💡A spring-loaded cart releasing two carts in opposite directions, a firework detonating, or a rocket separating from its booster are all explosions in this sense — internal forces doing the work, no outside force required to make it happen.

Explore all three scenarios from this unit's header art directly below — elastic rebound, perfectly inelastic collision, and explosion — and see how each cart's momentum changes from before to after.

The three scenarios from this unit's header art. Pick one and watch how momentum for each cart changes from before to after.

BEFORE
p₁ᵢ = (3)(6) = 18.0 kg·m/sp₂ᵢ = (5)(0) = 0.0 kg·m/s
AFTER
p₁f = (3)(-2.25) = -6.8 kg·m/sp₂f = (5)(3.75) = 18.8 kg·m/s
Total before: 18.0 kg·m/s  ·  Total after: 12.0 kg·m/s

Cart 1 approaches, bounces backward; cart 2 moves off in the direction cart 1 came from. Internal forces acted only during the brief contact.

ExampleGuided Example — Classifying an Interaction

Two carts sit motionless, connected by a compressed spring. When released, the spring pushes them apart, one moving left and one moving right. Is this a collision or an explosion, and why?

Step 1Identify the direction of the internal force
The spring's internal force pushes the two carts apart, away from each other — not together.
← Back to Lesson 4.1Ready for 4.2? Change in Momentum and Impulse connects force over time directly to momentum's change.