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
• Two-Dimensional Collisions & Explosions15 min · 3 objectivesConserve momentum independently along each axis in a two-dimensional collision · Recombine component momenta into a resultant magnitude and direction · Analyze an explosion as a collision run backward, starting from zero momentum
• Choosing Between Momentum and Energy15 min · 3 objectivesState the distinct conditions under which momentum and mechanical energy are conserved · Identify which conservation law a given problem allows and which it forbids · Compute the mechanical energy lost in a perfectly inelastic collision

Formulas in Unit 4

Every term in Unit 4

All 44 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.

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.

System choice

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

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.

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.

Momentum

p = mv, a vector pointing along the velocity, in kg·m/s. A heavy slow object and a light fast one can have equal momentum and very different kinetic energy.

Impulse-momentum theorem

J = Δp = mΔv. The bridge between force and velocity change, and the reason airbags work: extend Δt to reduce F for the same Δp.

Conservation of momentum

Total momentum is constant when the net EXTERNAL impulse on the system is zero. The qualifier is the whole content of the statement.

Why momentum is conserved in collisions

The forces between colliding objects are internal and equal-and-opposite by Newton's third law, so they cancel over the system.

Elastic collision

Both momentum and kinetic energy are conserved. Objects bounce apart, and this is the only collision type where kinetic energy is a usable equation.

Perfectly inelastic collision

The objects stick together and move with one common velocity. The maximum kinetic energy loss consistent with momentum conservation.

Solving a perfectly inelastic collision

m₁v₁ + m₂v₂ = (m₁ + m₂)v′. One equation, one unknown — and using kinetic energy here is the standard mistake, since it is not conserved.

Explosion problems

Momentum before is often zero, so the fragments' momenta must sum to zero afterward — equal and opposite for two pieces.

Two-dimensional collisions

Conserve momentum in each direction separately. The x and y equations are independent, and combining them into one is what goes wrong.

Center-of-mass velocity in a collision

Unchanged by the collision, since the collision forces are internal. A useful check on any collision answer.

Momentum vs kinetic energy

Momentum is a vector proportional to v; kinetic energy is a scalar proportional to v². Momentum is conserved in every collision, kinetic energy only in elastic ones.

Reading a force-time graph

Area gives impulse, and therefore the change in momentum. Peak force matters for whether something breaks; area matters for how the motion changes.

Why a longer collision time reduces force

For a fixed Δp, F = Δp/Δt. Crumple zones, catching a ball with give, and landing with bent knees all increase Δt.

Momentum in a system with external forces

Momentum changes at a rate equal to the net external force: ΣF = Δp/Δt. This is Newton's second law in its more general form.

Momentum conservation is per-axis

In two dimensions it is two independent equations, x and y. Solve each separately and recombine only at the end — speeds never add arithmetically.

An explosion is a collision run backward

Total momentum starts at zero and must end at zero, so two fragments fly apart with equal momentum magnitudes. The lighter piece moves faster in exact proportion.

K = p²/2m

Substituting p = mv into ½mv². With equal momentum the lighter object has more kinetic energy; with equal kinetic energy the heavier object has more momentum.

The condition for momentum conservation

Zero NET EXTERNAL force. Internal forces are third-law pairs and cancel, which is why collisions conserve momentum even though each object's momentum changes wildly.

Why collisions can ignore friction

The collision lasts milliseconds, so the friction impulse f·Δt is negligible against the collision impulse. Over a longer interval friction would break conservation.

Perfectly inelastic loses the maximum

Sticking together leaves the objects with the least kinetic energy momentum conservation permits. Maximum loss, but never total loss unless the initial momentum was zero.

Momentum vs kinetic energy conservation

Different conditions. Momentum needs zero net external force; mechanical energy needs no non-conservative work. A sticking collision satisfies the first and violates the second.

Why impulse spreads force over time

FΔt = Δp with Δp fixed. Lengthening the contact time lowers the force — airbags, crumple zones, bending your knees on landing.

Impulse from a force–time graph

The area under the curve, which is how a varying collision force is handled. The average force is that area divided by the duration.

Recombining component momenta

|p| = √(pₓ² + pᵧ²) and θ = tan⁻¹(pᵧ/pₓ). Adding magnitudes instead is the standard two-dimensional error.

Momentum is a vector

It has the direction of the velocity, so momenta are combined by components. Two equal masses moving in opposite directions at equal speed have zero total momentum, not double.

Impulse and momentum share units

Both are kg m/s, equivalently N s. If an impulse comes out in newtons or joules, a time or a distance has gone missing.

Newton's second law as dp/dt

F_net = dp/dt is the general form; F = ma is the special case for constant mass. The general form is what handles a rocket or a chain being lifted.

Relative speed reverses in an elastic collision

For a one-dimensional elastic collision, the speed of approach equals the speed of separation. It is a fast substitute for the kinetic energy equation.

A system at rest has zero total momentum

That stays true after an internal event. An explosion from rest sends fragments out with momenta that must cancel to zero when added as vectors.

Impulse is the area, not the peak force

A tall narrow force spike and a low broad one can deliver identical impulse. Reading the maximum off a force-time graph answers a different question.

Equal speeds, different momenta

A truck and a bicycle at the same speed carry very different momentum because momentum scales with mass. Speed alone never settles a momentum comparison.

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.
• Draw and label a before/after sketch with your axes marked before writing anything. On two-dimensional momentum free-responses, the two component equations are separate rubric points, and setting them up correctly earns credit even if the arithmetic goes wrong.
• When a free-response has a collision followed by a swing, slide or rise, it almost always wants momentum for the collision and energy for what follows. Using energy through the collision itself is the single most common way students lose the whole question.

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