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.

Options

A. `2/(sqrt2+1)`
B. `1/(sqrt2+1)`
C. `2/(sqrt2-1)`
D. `1/(sqrt2-1)`

Answer

A

Explanation

Let the stone is projected with a velocity `u` at an angle `theta` with the horizontal, we have
`(2h)=((u sin theta)^(2))/(2g) or u sin theta=2sqrtgh`
Suppose `t` is the time taken by the stone to reach the height `h` above the ground.Then
`h=u sin theta t-1/2g t^(2) or (g t^(2))/2-u sin thetat+h=0`
As we have `u sin theta =2sqrt(gh)`
`:. (g t)^(2)/2-2sqrt(gh)t+h=0`
Solving above equation for `t`, we get
`t=(2sqrt(gh)+-sqrt((2sqrt(gh))^(2)-4xxg/hh))/(2xxg/2)&t_(2)sqrt((2h)/g)(sqrt2+1)`
Where `t_(1)` and `t_(2)` correspond to `P` and `Q` in the figure.Suppose `v` is the horizontal velocity of the bird. Then `PQ=vt_(2)`.