Linear Momentum

What this unit covers

The topics below follow the published Physics 1 course framework for Unit 4. This unit is worth 10–15% of the exam, so budget your time against that rather than against how long the unit takes to teach.

Lessons in this unit

• Momentum & Impulse13 min · 3 objectivesCompute momentum as p = mv and treat it as a vector · Apply the impulse–momentum theorem, J = FΔt = Δp · Explain how extending the contact time reduces the force
• Conservation of Momentum14 min · 3 objectivesState the conservation of momentum for an isolated system · Apply it to recoil and explosion problems · Solve for an unknown velocity after an interaction
• Collisions13 min · 3 objectivesDistinguish elastic from inelastic collisions · Solve a perfectly inelastic collision where objects stick together · Recognize that kinetic energy is conserved only in elastic collisions
• Center of Mass12 min · 3 objectivesDefine the center of mass as the mass-weighted average position · Locate the center of mass of a two-object system · Explain how the center of mass moves when no external force acts

Formulas in Unit 4

Every term in Unit 4

All 18 terms we publish for Linear Momentum, with definitions. Reading them through is the fastest way to find the ones you cannot define — then drill those in cram mode until you can produce them without the prompt.

Center of mass

The mass-weighted average position. With no net external force, the center of mass of a system moves at constant velocity regardless of internal events.

Momentum

p = mv, a vector in the direction of velocity. A heavy slow object can carry the same momentum as a light fast one.

Impulse

J = FΔt = Δp. Extending the collision time lowers the force for the same momentum change — the principle behind airbags and crumple zones.

Impulse from a graph

The area under a force-time graph, which is how impulse is found when force varies during the collision.

Conservation of momentum

Total momentum is conserved when no net external force acts on the system. Internal forces cancel by Newton's third law.

System choice

Defining the system determines which forces are internal. Momentum conservation is a statement about a chosen system, not about objects.

Elastic collision

Both momentum and kinetic energy are conserved. Objects bounce apart, and relative speed of approach equals relative speed of separation.

Inelastic collision

Momentum is conserved but kinetic energy is not — some becomes thermal and deformation energy. Perfectly inelastic means the objects move together afterward.

Explosions

Momentum conservation run backward: an initially stationary system separates with equal and opposite momenta, so the lighter fragment moves faster.

Two-dimensional collisions

Momentum is conserved independently in x and y, so each direction gives its own equation.

Momentum vs kinetic energy

Momentum is a vector and always conserved in an isolated system; kinetic energy is a scalar and conserved only in elastic collisions.

Why airbags work

They extend Δt for the same Δp, and since FΔt = Δp, a longer collision means a smaller peak force on the body.

Recoil

A gun and bullet start with zero total momentum, so their final momenta are equal and opposite. The heavier gun recoils slowly.

Ballistic pendulum

Momentum is conserved during the embedding collision, then mechanical energy is conserved during the swing. Using energy for the collision is the classic error.

Identifying whether momentum is conserved

Ask whether any external force acts during the interaction. Over a short collision, gravity and friction are usually negligible compared with the impact force.

Perfectly inelastic collision speed

v = (m₁v₁ + m₂v₂)/(m₁ + m₂), since the objects move together afterward.

Kinetic energy lost in a collision

Initial KE minus final KE. It goes into deformation, sound and heat, and is largest in a perfectly inelastic collision.

Center-of-mass velocity

v_cm = Σmv/Σm. It is unchanged by any collision, which is a quick check on a collision answer.

What examiners penalize here

• When asked why a safety feature reduces force, argue from J = FΔt with Δp fixed. The impulse (momentum change) is set by the collision, so a longer Δt forces a smaller F. Name the theorem to earn the reasoning point.
• For any "at rest, then flies apart" problem, set total momentum equal to zero. The pieces carry equal and opposite momenta, so the mass ratio is the inverse of the speed ratio — heavier means slower.
• Two equations, two situations: momentum conservation works for *every* collision, but the kinetic-energy conservation equation is only valid for collisions the problem calls elastic. Never assume energy is conserved unless told so.
• Internal forces never move the center of mass. Whenever a problem shows an object breaking apart or flexing with no external force, the CM keeps whatever motion it had — a fast way to reason about explosions and recoil.

Practice Physics 1

Our practice bank is drawn from across the whole course rather than filtered to one unit, which is closer to how the exam asks anyway — it will not tell you which unit a question is testing.

Questions about this unit

All 8 units of AP Physics 1: Algebra-Based

1. Unit 1 · Kinematics
2. Unit 2 · Force and Translational Dynamics
3. Unit 3 · Work, Energy, and Power
4. Unit 4 · Linear Momentum
5. Unit 5 · Torque and Rotational Dynamics
6. Unit 6 · Energy and Momentum of Rotating Systems
7. Unit 7 · Oscillations
8. Unit 8 · Fluids