Exercise 8

Suppose you drop a golf ball onto a hard surface from a height $h$. The collision with the ground causes the ball to lose energy and so it will not bounce back to its original height. The ball will then fall again to the ground, bounce back up, and continue. Assume that at each bounce the ball rises back to a height $\frac{3}{4}$ of the height from which it dropped. Let $h_n$ be the height of the ball on the $n$th bounce, with $h_0 = h$. In this exercise we will determine the distance traveled by the ball and the time it takes to travel that distance.

1. Determine a formula for $h_1$ in terms of $h$.

2. Determine a formula for $h_2$ in terms of $h$.

3. Determine a formula for $h_3$ in terms of $h$.

4. Determine a formula for $h_n$ in terms of $h$.

5. Write an infinite series that represents the total distance traveled by the ball. Then determine the value of this series.

6. Next, let's determine the total amount of time the ball is in the air.

   1. When the ball is dropped from a height $H$, if we assume the only force acting on it is the acceleration due to gravity, then the height of the ball at time $t$ is given by $$\begin{gathered} H - \frac{1}{2}gt^2 \end{gathered}$$. Use this formula to determine the time it takes for the ball to hit the ground after being dropped from height $H$.

   2. Use your work in the preceding item, along with that in (a)-(e) above to determine the total amount of time the ball is in the air.