What is the value of n?

Question

A ball is dropped from an height of 1m. After hitting the floor the ball bounces back and reaches half of its previous height. After hitting the ball n times the potential energy of the ball becomes one-eighth of its initial potential energy. What is the value of n?

Answer 1

You can figure this out by iteration.  Start with initial drop from a height H.

 

Bounce 1:

H → H₁ = H/2

 

Bounce 2:

H₁ → H₂ = H₁/2 = H/2²

 

Bounce 3:

H₂ → H₃ = H₂/2 = H/2³

 

Do you see the pattern here?

 

Bounce k:

H_k → H_k-1/2 = H/2^k

 

If it reached 1/8 of its initial potential energy, then it reached H/8 of its initial height.

 

H/8 = H/2ⁿ

8 = 2ⁿ

n = 3