A tennis ball is dropped from a height of 7 ft. If the ball rebounds 2/3 of its height on each bounce, how far will the ball travel before coming to rest? (Find the total distance that the ball will travel before it stops bouncing.)

Question

i need help i don't know which formula to use to solve this problem.

Answer 1

in theory, the ball will never come to rest, since its bounce height is never zero.

But, you can see the height of the nth bounce is 7(2/3)^n

So, since the distance traveled is both up and down, the sequence is

7 + 2*7(2/3) + 2*7(2/3)^2 + ...
so, if r=2/3,
= 7 + 14(r+r^2+...)
After n bounces the sum is
7 + 14((1-r^n)/(1-r) - 1)
and as n->infinity, that leaves
7 + 14(3/2 - 1) = 7+7 = 14

I think ...