Elastic and inelastic collisions

Definition of a collision

A collision is a brief interaction between two bodies, during which the internal contact forces are much greater than the external forces (weight, friction), which can therefore be neglected for the duration of the collision. The total momentum of the system is therefore conserved:

m1v1 + m2v2 = m1v1' + m2v2'

Perfectly inelastic (or soft) collision

The two bodies remain stuck together after the collision and now have a single common velocity v’:

m1v1 + m2v2 = (m1+m2)*v’

Kinetic energy is NOT conserved in this case: some of it is dissipated in the form of deformation, heat or sound.

Elastic collision

The two bodies separate after the collision and the total kinetic energy IS conserved, in addition to the momentum:

(1/2)m1v1^2 + (1/2)m2v2^2 = (1/2)m1v1'^2 + (1/2)m2v2'^2

This system of two equations (conservation of momentum and kinetic energy) allows us to determine the two unknown final velocities.

Special case: equal masses, 1D elastic collision

If m₁ = m₂ and v₂ = 0 (stationary target), we find v₁' = 0 and v₂' = v₁: the velocities are simply swapped. This model accurately describes the collision of two identical billiard balls.

Coefficient of restitution

e = -(v1' - v2') / (v1 - v2), where e = 1 for a perfectly elastic collision and e = 0 for a perfectly inelastic collision. In practice, 0 < e < 1.

Common pitfall

Never assume that the conservation of kinetic energy holds true by default during a collision: it is true ONLY for elastic collisions, whereas the conservation of momentum holds true for any collision within an isolated system.