A lecturer driving his car of mass $m_{1}$ along the flat at speed $U_{1}$ accidentally collides with a stationary vehicle of mass $m_{2}$. As both vehicles are old and very solidly built, neither suffers damage in the collision: they simply bounce elastically off each other in a straight line. Determine how both vehicles are moving after the collision if neither driver applied their brakes. State any assumptions made and consider all possible values of the mass ratio $R=m_{1} / m_{2}$. You may neglect friction and other such losses.

An undergraduate drives into a rigid rock wall at speed $V$. The undergraduate's car of mass $M$ is modern and has a crumple zone of length $L$ at its front. As this zone crumples upon impact, it exerts a net force $F=(L-y)^{-1 / 2}$ on the car, where $y$ is the amount the zone has crumpled. Determine the value of $y$ at the point the car stops moving forwards as a function of $V$, where $V<2 L^{\frac{1}{4}} / M^{\frac{1}{2}}$.