4.II.12C

In an experiment a ball of mass $m$ is released from a height $h_{0}$ above a flat, horizontal plate. Assuming the gravitational acceleration $g$ is constant and the ball falls through a vacuum, find the speed $u_{0}$ of the ball on impact.

Determine the speed $u_{1}$ at which the ball rebounds if the coefficient of restitution for the collision is $\gamma$. What fraction of the impact energy is dissipated during the collision? Determine also the maximum height $h_{n}$ the ball reaches after the $n^{\text {th }}$bounce, and the time $T_{n}$ between the $n^{t h}$ and $(n+1)^{t h}$ bounce. What is the total distance travelled by the ball before it comes to rest if $\gamma<1$ ?

If the experiment is repeated in an atmosphere then the ball experiences a drag force $D=-\alpha|u| u$, where $\alpha$ is a dimensional constant and $u$ the instantaneous velocity of the ball. Write down and solve the modified equation for $u(t)$ before the ball first hits the plate.

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