Question

Suppose a ball is dropped from a height of 14 feet. Each time it drops h feet, it rebounds 0.91h feet. Find the total distance traveled by the ball. Round your answer to two decimal places.

Answer 1

h - fell
0.91 h - went up
0.91 h - came back down
0.91 x (0.91h) - went up
0.91 x (0.91h) - came back down
..... and so on

therefore we will consider it in 2 different motions, and then add them as we need the total distance of travel by ball
going down
then going up

for going down
h, 0.91h, 0.91(0.91)h, ...

a geometric progression, where the common ratio = 0.91
and the value of first term or h = 14 given to us ..

now we dont know until when it will continue, as this will go on, bouncing down infinitely ; we will therefore find out the sum to infinity.
sum to infinity of geometric sequence = (first term) / (1 - common ratio) = 14/(1-0.91) = 155.55 feet

now going up we will consider

0.91h, 0.91(0.91h), ... and so on
geometric sequence

here again the common ratio = 0.91
now here the first term is 0.91h = 0.91x(14)
sum to infinity of geometric sequence = (first term) / (1 - common ratio) = [(0.91) x 14]/(1-0.91) = 141.55 feet

total distance is by adding both = 155.55+141.55 = 297.1 feet

Hence the total distance travelled by the ball is

297.1 ft