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#

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