
Arturo O. answered 12/23/17
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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:
Hk → Hk-1/2 = H/2k
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