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

`sqrt(3) - 1`

Explanation

`x_2 = u cos theta t, x_2 - x_1 = v - t`
`rArr x_1 + vt = u cos theta t`
`h = x tan theta - (gx^2)/(2 u^2 cos^2 theta), (3 h)/(2) = (u^2 sin^2 theta)/(2 g)`
Simplify to get `x^2 - 6 h cot theta x + 6h^2 cot^2 theta = 0`
`x = (6 h cot theta +- sqrt(12) h cot theta)/(2) = 3 h cot theta +- sqrt(3) h cot theta`
`x_1 = 3 h omega "to" -sqrt(3) h and x_2 = 3h cot theta + sqrt(3) h`
`v = (x_2 - x_1)/(t) = (2 sqrt(3) h cot theta)/(t)`
`u cos theta = (x_2)/(t) = ((3 + sqrt(3))h cot theta)/(t)`
`(v)/(u cos theta) = (2 sqrt(3))/(3 + sqrt(3)) = (2sqrt(3)(3 - sqrt(3)))/(9 - 3) = sqrt(3) -1`.
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