Distance Traveled Help

Question

A ball drops from a height of 13 feet. Each time it hits the ground, it bounces

up 40 percents of the height it falls. Assume it goes on forever, find the total

distance it travels.

Answer 1

The first bounce, a₁, is 10.4 feet, 5.2' up, 5.2' down

S = 10.4 / (1 - 0.40)

S = 10.4 / 0.60

S = 17.3333 (rounded down)

Total distance add 13 foot drop.

Answer 2

sounds suspiciously like a geometric series problem

general formula is:

Sn = a1(1 -r^n)/(1-r)

r=40% = 0.4

a1 =13

but you want an infinite series

S =a/(1-r) = 13/(1-.4)= 13/.6 = 130/6 = 21 2/3 feet

but that is only about very roughly half of the total distance

You want S = 13 + 2a/(1-r) = with a=13(.4) = 5.2, r=.4

Sum = 13 + 10.4/.6 = 13 + 17 1/3 = 30 1/3 feet

you need to double the terms from 5.2 on, as it goes down same distance it went up for every bounce, then add 13 which only happens once on its way down