A ball drops from a ceiling of a room and after rebounding twice from the floor reaches a height equal to half that of the ceiling. Show that the coefficient of restitution is `sqrt(1/2)`

Explanation

Let h = height of ceiling.
Speed before first impact = `sqrt(2gh)`
Speed after first impact = `e sqrt(2gh)`
The ball comes back for second impact. Before second impact, speed = `e sqrt (2gh)`
After second impact, speed = `e^(2) sqr(2gh)`
The speed is the initial speed u for upward motion after second impact,
Heigh attained ` = u^(2)/(2g)`
`Rightarrow h/2 = [e^(2) sqrt(2gh)]^(2)/2g`
`Rightarrow h/2 = (e^(4)(2gh))/(2g) Rightarrow e = 1/4sqrt 2`