Geometric Question

Question

A ball drops from a height of 25 feet. Each time it hits the ground, it bounces

up 65 percents of the height it falls. Assume it goes on forever, find the total

distance it travels.

Answer 1

The first bounce is 32.5', (2)(25)(0.65)

a₁ = 32.5

r = 0.65

S = a₁ / (1 - r)

Total distance is 25 + S

Answer 2

The simple answer is that it's a geometric sequence. The sum can be found like so:

25*[1+.65 + 65² +.65³ +.65⁴ +.65⁵ +...] the down portion, plus

25*[.65 + .65² +.65³ +.65⁴ +.65⁵ +...] the up portion

---------------------------------------------

25 + 50*[.65 + .65² +.65³ +.65⁴ +.65⁵ +...]

25 + 50*(.65/(1-.65))

≅ 117.9 feet

But it's all so much more interesting than that! I will post an animation of this result momentarily.

edit: here is the animation: https://drive.google.com/file/d/1mgPSJR8Yjt4aIvicEfGojIv4Mkdi0v5h/view?usp=sharing