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Question:

A ball is bouncing down a flight of stairs. The coefficient of restitution is e. The height of each step is d and the ball descends one step each bounce. After each bounce it rebounds to a height h above the next lower step. The height is large compared with the width of step so that the impacts are effectively head-on. Find the relationship between h and d.

h=√(d(1−e²))

h=d1+e2

h=d(1−e²)

h=d1+e

Solution:

Since v²−u²=2as
When falling down, 2ah=v₁²
When bouncing back up, 2a(h−d)=v₂²
Thus the coefficient of restitution e=v₂/v₁=√((h−d)/h) => h=d/(1−e²)