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Physics Unit 4: Momentum

Momentum is mass in motion. In any collision with no outside force, total momentum before equals total momentum after.

Covers: Momentum, impulse, conservation of momentum, elastic and inelastic collisions

Study guide

Momentum is mass in motion: p = mv, a vector measured in kg·m/s. A heavy truck and a fast bullet can both have large momentum.

Impulse is the change in momentum, J = Δp, and it equals force times the time the force acts: J = FΔt. For the same change in momentum, a longer stopping time means a smaller force. That's why airbags, padding, crumple zones and bending your knees when you land reduce injuries. On a force-time graph, impulse is the area under the curve.

The law of conservation of momentum says that when no external net force acts on a system, its total momentum stays constant. In collisions and explosions, the total momentum before equals the total momentum after: m₁v₁ + m₂v₂ = m₁v₁′ + m₂v₂′. Choose a positive direction and use negative signs for objects moving the other way. For an explosion or recoil starting from rest, the total momentum stays zero, so the pieces move in opposite directions with equal and opposite momenta.

Collisions are classified by kinetic energy. In an elastic collision, both momentum and kinetic energy are conserved, as in billiard balls (approximately) and atomic collisions. In an inelastic collision, momentum is conserved but some kinetic energy becomes heat, sound or deformation. In a perfectly inelastic collision, the objects stick together and share a final velocity: v′ = (m₁v₁ + m₂v₂)/(m₁ + m₂). Momentum is conserved in all three types.

In two dimensions, conserve momentum separately in the x and y directions.

The center of mass of a system moves with velocity v_cm = (total momentum) ÷ (total mass). With no external force, v_cm stays constant before, during and after any collision or explosion, whether the collision is elastic or the objects stick. Its position-time graph is one straight line. Forces that objects in a system exert on each other, like friction between a block and the cart it rides on, are internal and don't change the system's total momentum. By Newton's third law, every internal interaction gives the objects equal and opposite changes in momentum. Only a net external force changes the total.

In a one-dimensional elastic collision, the objects separate at the same relative speed at which they approached. When a moving object hits an identical one at rest elastically, it stops and the other moves off with its velocity.

Key ideas

• Momentum p = mv is a vector.
• Impulse J = FΔt = Δp.
• Total momentum is conserved when no external net force acts.
• Elastic collisions also conserve kinetic energy; inelastic ones don't.
• Perfectly inelastic: objects stick together and share a final velocity.

Formulas and rules

Common mistakes

• Forgetting momentum's direction (signs).
• Assuming kinetic energy is conserved in every collision.

Words to know

Momentum

p = mv

Impulse

FΔt, equal to the change in momentum.

Conservation of momentum

Total momentum stays constant when no external net force acts.

Perfectly inelastic collision

Objects stick together.

Impulse and stopping time

A longer stopping time means a smaller force for the same impulse (why airbags work).

Perfectly inelastic final velocity

v′ = (m₁v₁ + m₂v₂)/(m₁ + m₂)

Elastic collision

Momentum and kinetic energy are both conserved.

Inelastic collision

Momentum is conserved; some kinetic energy is lost.

Recoil

Objects pushed apart from rest move opposite with equal momenta.

Force-time graph

The area under it is the impulse.

System (momentum)

The objects you track; external forces change its total momentum.

2D momentum conservation

Conserve x- and y-components separately.

Sample practice questions

19 more questions like these, from level 1 to 5, are in the practice set.