Question

A stone is projected from a point on the ground in such a direction so as to hit a bird on the top of a telegraph post of height $h$, and then attain a height $2h$ above the ground. If, at the instant of projection, the bird were to fly away horizontal with a uniform speed, find the ratio of the horizontal velocities of the bird and the stone if the stone still hits the bird.

Answer 1

Maximum height of the projectile is given by the expression

$h_{max}=\frac{v_0^2{\sin }^2\theta }{2g}$

$\Rightarrow 2h=\frac{v_0^2{\sin }^2\theta }{2g}\Rightarrow v_0=\frac{2\sqrt{gh}}{\sin \theta }$...................(i)

$\because y=v_0\sin \theta t-\frac{1}{2}g{t}^2$

$\Rightarrow h=\frac{2\sqrt{gh}}{\sin \theta }\sin \theta t-\frac{1}{2}g{t}^2\Rightarrow g{t}^2-4\sqrt{ght}+2h=0$

$t=\frac{4\sqrt{gh}\pm \sqrt{16gh-8gh}}{2g}$

$=\frac{4\sqrt{gh}\pm 2\sqrt{2gh}}{2g}$

$\Rightarrow t_1=\sqrt{\frac{h}{g}}(2-2\sqrt{2})$ and $t_2=\sqrt{\frac{h}{g}}(2+\sqrt{2})$

At these two times , the projectile is at the same height as the bird.

let $v$ be the speed of the bird , for bird to be hitted Therefore,

$v{t}_2=v_a\cos \theta (t_2-t_1)$

$\Rightarrow \frac{v}{v_0\cos \theta }=\frac{t_2-t_1}{t_2}=\frac{2}{\sqrt{2}+1}$