Putnam 1941 B5

A car is being driven so that its wheels, all of radius $a$ feet, have an angular velocity of $\omega$ radians per second. A particle is thrown off from the tire of one of these wheels, where it is supposed that $a \omega^{2} >g$ . Neglecting the resistance of the air, show that the maximum height above the roadway which the particle can reach is $$\frac{(a \omega+g \omega^{-1})^{2}}{2g}.$$