A stone is projected from the point on the ground in such a direction so on to hit a bird on the top of a telegraph post of height h and then attain maximum height 2h above the ground. If at the instant of projection bird were to fly away horizontally with uniform speed $v=2(\sqrt{2}-1)$~m/s. Find the horizontal velocity of the stone if the stone still hits the bird while descending.

The situtaion is shown in figure let $\theta$ be the angle of projection and $u$ the speed of projection
Maximum height $MN=2h$
$MN=2h=\frac{u^2{\sin }^2\theta }{2g}$
$u\sin \theta =2\sqrt{gh}$
Let $t$ be the time taken by the stone to attain the vertical height $h$ above the ground.
$h=\left(u\sin \theta \right)t-\frac{1}{2}g{t}^2$
$\Rightarrow t=\frac{u\sin \theta }{g}\pm \sqrt{\frac{u^2{\sin }^2\theta }{g}-\frac{2h}{g}}$
$t=sqrt\frac{4h}{g}\pm \sqrt{\frac{2h}{g}}$
$\Rightarrow t_1=\sqrt{\frac{4h}{g}}-\sqrt{\frac{2h}{g}}{t}_2=\sqrt{\frac{4h}{g}}+\sqrt{\frac{2h}{g}}$

Where $t_1$ and $t_2$ one time to reach A and B respectively. If v is the horizontal velocity of brid


$\Rightarrow \frac{v}{u\cos \theta }=\frac{2}{\sqrt{2}+1}$
$\Rightarrow u\cos \theta =1m/s$