Chapter 7, Problem 36P

To determine

To Find:Number of rebounds that the ball will make before losing $90\%$ of its energy.

Explanation of Solution

Given:

The height from which the ball is dropped, $h_1=1.50\text{m}$

The height to which the ball rebounds, $h_2=1.20\text{m}$

Percentage of energy lost by the ball $=90\%$

Calculations:

Fraction of energy retained after first rebound is $=\frac{h_2}{h_1}=\frac{1.20}{1.50}=0.8$ .

After n rebounds, amount of energy retained:

  $\begin{array}{l}0.1={\left(0.8\right)}^n\\ \log \left(0.1\right)=n\log \left(0.8\right)\\ n=\frac{\log \left(0.1\right)}{\log \left(0.8\right)}=10.31\approx 10\end{array}$

Conclusion: 

Thus, the ball would make about 10 rebounds before it loses $90\%$ of its energy.