Elastic versus perfectly inelastic

Problem

Elastic versus perfectly inelastic. Explain in a few lines the difference between an elastic collision and a perfectly inelastic collision, stating which conservation law holds in each case and where the missing energy goes in the inelastic one.

Solution

In both types of collision, total momentum is conserved (the collision forces are internal): so in both cases the velocity of the centre of mass remains unchanged.

• Elastic collision: kinetic energy is also conserved. The bodies bounce apart; $E_\text{cin,int}$ before and after is the same.
• Perfectly inelastic collision: the bodies remain stuck together and only momentum is conserved. Part of the kinetic energy — precisely all of the $E_\text{cin,int}$, the energy in the CM frame — is lost.

$\ev{\text{elastic: } \vec p, E_\text{cin} \text{ conserved} \quad|\quad \text{fully inelastic: only } \vec p}$

The missing energy in the inelastic case does not vanish: it is converted into heat, permanent deformation and sound.

Links

Topics: Momentum and collisions Concepts: Elastic collision · Inelastic collision · Conservation of momentum