A stone is projected from the point of a ground in such a direction so as to hit a bird on the top oftelegraph post of height h and then attain the maximum height 2h above the ground. If at the instant of projection, the bird were to fly away horizontally with a unifrom speed. Find the ratio between the horizontal velocities of the bird and stone, if the stone still his the bird while ascending.

A stone is projected from the point of a ground in such a direction so

1.	A stone is projected from the point of a ground in such a direction so as to hit a bird on the top oftelegraph post of height h and then attain the maximum height 2h above the ground. If at the instant of projection, the bird were to fly away horizontally with a unifrom speed. Find the ratio between the horizontal velocities of the bird and stone, if the stone still his the bird while ascending.
Answer» Solution :Let the stone is projected with a velocity u at an angle `theta`with the horizontal, we have `(2h)=((usin theta)^(2))/(2g)` or `u SIN theta=2sqrt(gh)` Suppose t is the time taken by the stone to reach the height h above the ground. Then `h=u sin theta t-1/2"GT"^(2)` (or) `("gt"^(2))/2- u sin theta+h=0` As we have `u sin theta =2sqrt(gh)"" :.("gt"^(2))/2-2sqrt(gh)t+h=0` Solving above equation for t we get `t=(2sqrt(TH)+-sqrt((2sqrt(gh))^(2)-4xxg/2h))/(2xxg/2)=(2sqrt(gh)+-sqrt(2gh))/g` Let `t_(1)=sqrt((2h)/g)(sqrt(2)-1)` and `t_(2)=sqrt((2h)/g)(sqrt(2)+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)` For the motion of stone `PQ=u cos theta (t_(2)-t_(1))"":.vt_(2)=ucos theta(t_(2)-t_(1))` which gives `v/(u cos theta)=((t_(2)-t_(1))/(t_(2)))=(sqrt((2gh)/g)(sqrt(2)+1)-(sqrt(2)-1))/(sqrt((2h)/g)(sqrt(2)+1))=2/(sqrt(2)+1)`