Dynamics | Part IA, 2005

In an experiment a ball of mass mm is released from a height h0h_{0} above a flat, horizontal plate. Assuming the gravitational acceleration gg is constant and the ball falls through a vacuum, find the speed u0u_{0} of the ball on impact.

Determine the speed u1u_{1} at which the ball rebounds if the coefficient of restitution for the collision is γ\gamma. What fraction of the impact energy is dissipated during the collision? Determine also the maximum height hnh_{n} the ball reaches after the nth n^{\text {th }}bounce, and the time TnT_{n} between the nthn^{t h} and (n+1)th(n+1)^{t h} bounce. What is the total distance travelled by the ball before it comes to rest if γ<1\gamma<1 ?

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

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