4.4 Momentum in Elastic and Inelastic Collisions

Momentum is always conserved

In any collision with no external net force, total linear momentum of the system is conserved: p_before = p_after. This holds whether the collision is elastic or inelastic. Kinetic energy conservation depends on the type of collision, so always start with momentum conservation, then check energy conditions.

Elastic collisions conserve KE too

In a perfectly elastic collision, both momentum AND kinetic energy are conserved. Objects bounce apart without deforming or losing energy to heat/sound. Using both conservation equations lets you solve for two unknown final velocities, given masses and initial velocities: v1f = ((m1-m2)/(m1+m2))v1i + (2m2/(m1+m2))v2i.

Inelastic collisions lose KE

In inelastic collisions, momentum is conserved but kinetic energy is not — some converts to heat, sound, or deformation. A perfectly inelastic collision is the extreme case: objects stick together and move with a common final velocity, found via m1v1i + m2v2i = (m1+m2)vf. Maximum KE is lost in this case.

Key terms

Elastic collision

A collision in which both momentum and kinetic energy are conserved; objects bounce apart without energy loss.

Inelastic collision

A collision in which momentum is conserved but kinetic energy is not; some energy converts to heat, sound, or deformation.

Perfectly inelastic collision

The extreme case of inelastic collision where objects stick together and move with a common final velocity; maximum KE is lost.

Elastic collision final velocity formula (1D)

v1f = ((m1−m2)/(m1+m2))v1i + (2m2/(m1+m2))v2i, and similarly for v2f by swapping subscripts.

Equal-mass elastic collision result

When m1 = m2 in a 1D elastic collision, the objects exchange velocities.

Try an AP-style question

A light ball bounces elastically off a very massive wall moving toward it at speed u. The ball approaches at v. What is its rebound speed?

1. v − 2u
2. v + 2u
3. v
4. v + u