Elastic and Inelastic Collisions

In an $\text{elastic}$ collision, total kinetic energy is conserved as well as momentum. Collisions between gas molecules are elastic, and so to a good approximation are collisions between snooker balls.

In an $\text{inelastic}$ collision, some kinetic energy is transferred to other forms — heat, sound, permanent deformation. Total energy is still conserved; it is only $\text{kinetic}$ energy that is not. A $\text{perfectly inelastic}$ collision is one where the objects stick together, losing the maximum kinetic energy consistent with momentum conservation.

The test is always the same: compute total kinetic energy before and after and compare.

Substituting $v=p/m$ into $E_k=\frac{1}{2}m{v}^2$ gives $E_k=\frac{p^2}{2m}$.

This form is very useful for explosions and recoils, where momenta are equal and opposite. With equal momentum magnitudes, kinetic energy is inversely proportional to mass — so the lighter fragment carries most of the energy.

Momentum is a vector, so in a glancing collision it is conserved $\text{separately in each of two perpendicular directions}$.

Resolve every velocity into components along chosen $x$ and $y$ axes, then write one conservation equation for each. Commonly the $x$-axis is along the initial motion, so the initial $y$-momentum is zero.

This gives two equations, enough to find two unknowns — typically a speed and an angle.

Particle physics relies on these conservation laws. If the momentum of detected particles after a decay does not balance, a particle must have carried the missing momentum undetected.

That reasoning led to the prediction of the neutrino: in beta decay the emitted electrons had a spread of energies and the momenta did not balance, which only made sense if a third, neutral particle was also emitted.

The same bookkeeping is applied to collisions in accelerators to identify the products and to infer particles that leave no track.