Answer
After 8 bounces, the ball will have lost 90% of its energy.
Work Step by Step
The ball starts at a height of 1.60 meters and rises to a height of 1.20 meters after the first rebound. We can find the fraction of potential energy that remains after the first rebound:
$\frac{mgh_2}{mgh_1} = \frac{1.20~m}{1.60~m} = 0.75$
We can assume that the ball loses 25% of its energy on each rebound. We can find the number of rebounds $n$ until only 10% of the original energy remains:
$(0.75)^n = 0.1$
$n = \frac{ln(0.1)}{ln(0.75)} = 8$
After 8 rebounds, the ball will have lost 90% of its energy.
