A superball that rebounds 3/10 of the height from which it fell on each bounce is dropped from 38 meters. ?

How high does it rebound, in meters, on the 8 th bounce?
How far does it travel, in meters, before coming to rest?

1 Answer
May 16, 2016

8 th bounce, height = 38(3/10)^8 distance traveled to rest = 70.5714

Explanation:

The sequence of heights after leaving is
38{1,3/10,(3/10)^2,(3/10)^3,...,(3/10)^8,...,(3/10)^n}
The space d traveled is given by
d=2times 38{1+3/10+(3/10)^2+(3/10)^3+...+(3/10)^n}-38
Now using the polynomial identity
(1-x^{n+1})/(1-x)=1+x+x^2+x^3+...+x^n
1+3/10+(3/10)^2+(3/10)^3+...+(3/10)^n= (1-(3/10)^{n+1})/(1-(3/10))
Supposing that n->infty,
then (3/10)^{n+1}->0 because (3/10)<1
So we get
1+3/10+(3/10)^2+(3/10)^3+...+(3/10)^n+...= 1/(1-(3/10)) = 10/7
Finally putting all together
d = 2 times 38 times 10/7 -38 = 70.5714