Q. 79 - Calculus

Question

Whenever a certain ball is dropped, it always rebounds to a height 60% of its original position. What is the total distance the ball travels before coming to rest when it is dropped from a height of 1 meter?

Step-by-Step Solution

Answer

The total distance traveled by the ball before coming to the rest when it is dropped from a height of 1 meter is $4\mathrm{meter}.$

Step 1. Given Information.

The ball after dropping rebounds to a height 60% of its original position. 

The given figure is

Step 2. Find the distance the ball travels before coming to rest when it is dropped from a height of 1 meter.

To find the total distance traveled by the ball, let's first find the distance traveled by the ball before coming to rest, each successive bounce has two times 60% of the height as it involves up and down of the ball,

$$\begin{gathered} d=1+2\times 1\times \left(0.60\right)+2\times 1\times {\left(0.60\right)}^2+... \\ d=1+2\times 1\times 0.60\left(1+0.6+{\left(0.6\right)}^2+...\right) \\ \mathrm{Use} \mathrm{the} \mathrm{formula} \mathrm{of} \mathrm{geometric} \mathrm{series}\left(\frac{\mathrm{a}}{1-\mathrm{r}}\right) \\ \mathrm{d}=1+2\times 1\times 0.60\left(\frac{1}{1-0.6}\right) \\ \mathrm{d}=1+1.2\left(\frac{1}{0.4}\right) \\ \mathrm{d}=1+3 \\ \mathrm{d}=4 \end{gathered}$$

Therefore, the total distance traveled by the ball before coming to the rest is $4\mathrm{m}.$