A ball drops from a height of 20 feet. Assume it goes on forever, find the total distance it travels.

Question

A ball drops from a height of 20 feet. Each time it hits the ground, it bounces up 40 percents of the height it fall. Assume it goes on forever, find the total distance it travels.

Answer 1

 

The 1st bounce:   Height: h₁ = h₀ (0.4) = 20 (0.4);  Distance travels: d₁ = 2h₁ = 40 (0.4)

The 2nd bounce:  Height: h₂ = h₁ (0.4) = 20(0.4)²; Distance travels: d₂ = 2h₂ = 40 (0.4)²

...

The n^th bounce:   Height: h_n = h_n-1(0.4) = 20(0.4)ⁿ; Distance travels: d_n = 2h_n = 40 (0.4)ⁿ

 

The total distance the ball travels is

 

D = h₀ + d₁ + d₂ +  ...

   = 20 + 40(0.4) + 40(0.4)² + ...

   = 20 + 40∑_n=1(0.4)ⁿ

   = 40 + 40∑_n=1(0.4)ⁿ − 20

   = 40∑_n=0(0.4)ⁿ − 20

   = 40[1/(1 − 0.4)] − 20

   = 140/3 ft