Past exam of the physics 1 paper of the University of Oxford / 2014 resit / 6 / Solution

Let the initial velocity be $\mathbf{v}$ and the two final velocities be $\mathbf{u},\mathbf{w}$. In the stated nonrelativistic elastic collision, conservation of momentum and conservation of energy giveSquaring the first equation and comparing it with the second yieldsIf both final velocities are nonzero, their directions are perpendicular. If one is zero, the moving particle retains the initial velocity and a scattering angle for the stationary particle is undefined. This includes the zero-deflection and head-on exchange limits rather than incorrectly assigning a direction to a zero vector. Equal masses and a stationary initial target are essential to the dot-product cancellation.