A stone is projected from the ground in such a direction so as to hit a bird on the top of a telegraph post of height h and attains the maximum height of 2h above the ground. If at the insatant of projection, the bird were to fly away horizontally with a uniform speed, find the ratio between the horizontal velocity of bird and the horizontal component of velocity of stone, if the stone hits the bird while descending.

Answer

`(2)/(sqrt(2 + 1))`.

Explanation

Maximum height of the projectile is given by the expression
`h_(max) = (v_0^2 sin^2 theta)/(2 g)`
`rArr 2h = (v_0^2 sin^2 theta)/(2 g) rArr v_0 = (2 sqrt(gh))/(sin theta)` …(i)
`because y = v_0 sin theta t - (1)/(2) "gt"^2`
`rArr h = (2 sqrt(gh))/(sin theta) sin theta t - (1)/(2) "gt"^2 rArr "gt"^2 - 4 sqrt(ght) + 2 h = 0`
`t = (4 sqrt(gh) +- sqrt(16 gh - 8 gh))/(2 g) = (4 sqrt(gh) +- 2 sqrt(2 gh))/(2 g)`
`rArr t_1 = sqrt((h)/(g)) (2 - sqrt(2)) and t_2 = sqrt((h)/(g)) (2 + sqrt(2))`
At these two times, the projectile is at the same height as the bird.
Let `v` be the speed of the bird, for to be hitted. Therefore,
`v t_2 = v_o cos theta (t_2 - t_1)`
`rArr (v)/(v_o cos theta) = (t_2 - t_1)/(t_2) = (2)/(sqrt(2 + 1))`.