Chapter 9.5, Problem 38PPSE

Solution

To find the total distance travelled by the ball.

The total distance travelled by the ball is 20 meters.

Given information:

A ball is dropped from 1 meter height and the ball bounces 95% of the previous height each time until it comes to rest.

Calculation:

The initial height of the ball is 1 meter, and the height of ball at the second instance will be $0.95\times 1$ meters. Proceeding in same way, the series can be written as $1+\left(0.95\right)+{\left(0.95\right)}^2+{\left(0.95\right)}^3+\mathrm{...}$ .

The geometric series has $a_1=1$ and $r=0.95$ . This information can be used to calculate the distance travelled by the ball until it comes to rest. So, the problem gives enough information to calculate the distance travelled by the ball.

Now, write the general term of the sequence.

  $\begin{array}{c}{a}_n=a_1{r}^{n-1}\\ =1\cdot {\left(0.95\right)}^{n-1}\\ ={\left(0.95\right)}^{n-1}\end{array}$

It can be assumed that the ball comes to rest for a larger value of $n$ . So, the formula for the sum of infinite geometric series can be used to calculate the total distance travelled by the ball.

Now, find the distance travelled.

  $\begin{array}{c}{S}_{\infty }=\frac{a_1}{1-r}\\ =\frac{1}{1-0.95}\\ =\frac{1}{0.05}\\ =20\end{array}$

Therefore, the total distance travelled by the ball is 20 meters.