A body falling freely from a given height `H` hits an inlclined plane in its path at a height `h`. As a result of this impact the direction of the velocity of the body becomes horizontal. For what value of `h//H`, the body will take the maximum time to reach the ground.

Explanation

`O` to `B`: time `t_(1)`, `H - h = (1)/(2)g t_(1)^(2) rArr t_(1) = sqrt((2(H - h))/(g))`
`B` to `C`: time `t_(2)`, `h = (1)/(2)g t_(2)^(2) rArr t_(2) = sqrt((2h)/(g))`
Total time `T = t_(1) + t_(2) = sqrt((2)/(g))[(H - h)^(1//2) + h^(1//2)]`
For `T` to be maximum
`(d T)/(d h) = 0`
`(1)/(2) (H - h)^(-1//2)(-1) + (1)/(2)h^(-1//2) = 0`
`H - h = h rArr (h)/(H) = (1)/(2)`