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
Cesareo R.
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#

Related questions
See all questions in Nth Term Test for Divergence of an Infinite Series
Impact of this question
2981 views around the world
You can reuse this answer
Creative Commons License