Kevin S. Huang

2008: Problem 24

Topic: Kinematics
Concepts: 1D inelastic collision, Free fall

Solution:

If a ball is launched upward with velocity $v$ , then it will reach the top in time

v-gt=0

t=\frac{v}{g}

The ball takes the same amount of time to come down so the time for the first bounce is

t_1=\frac{2v}{g}

For the second bounce, the velocity is modified by a factor of $r$ ,

t_2=\frac{2rv}{g}

An additional factor of $r$ is introduced for each subsequent bounce. Thus, the total time the ball bounces is

T=t_1+t_2+t_3+…=\frac{2v}{g}(1+r+r^2+…)=\frac{2v}{g}\frac{1}{1-r}

using the formula for the sum of an infinite geometric series. The answer is A.