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Derivation of Position-Velocity Relation by Graphical Method

A stone is pr...

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 $2 h$ 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.

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Solution

Maximum height of the projectile is given by the expression
$h_{m a x} = \frac{v_{0}^{2} {sin}^{2} θ}{2 g}$
$\Rightarrow 2 h = \frac{v_{0}^{2} {sin}^{2} θ}{2 g} \Rightarrow v_{0} = \frac{2 \sqrt{g h}}{sin θ}$ ...................(i)
$∵ y = v_{0} sin θ t - \frac{1}{2} g t^{2}$
$\Rightarrow h = \frac{2 \sqrt{g h}}{sin θ} sin θ t - \frac{1}{2} g t^{2} \Rightarrow g t^{2} - 4 \sqrt{g h t} + 2 h = 0$
$t = \frac{4 \sqrt{g h} \pm \sqrt{16 g h - 8 g h}}{2 g}$
$= \frac{4 \sqrt{g h} \pm 2 \sqrt{2 g h}}{2 g}$
$\Rightarrow t_{1} = \sqrt{\frac{h}{g}} (2 - 2 \sqrt{2})$ 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,
$v t_{2} = v_{a} cos θ (t_{2} - t_{1})$
$\Rightarrow \frac{v}{v_{0} cos θ} = \frac{t_{2} - t_{1}}{t_{2}} = \frac{2}{\sqrt{2} + 1}$

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