A stone is thrown in such a manner that it would just hit a bird at the top of a tree and afterwards reach a maximum height double that of the tree. If at the moment of throwing the stone the bird flies away horizontally with constant velocity and the stone hits the bird after some time. The ratio of the horizontal velocity of stone to that of the bird is $\frac{1}{n}+\frac{1}{\sqrt{n}}.$ Find $2n$.

If $4 \mathrm{s}$ be the time in which a projectile reaches a point $P$ of its path and $5 \mathrm{s}$ the time from $P$ till it reaches the horizontal plane passing through the point of projection. The height of $P$ above the horizontal plane (in $\mathrm{m}$ ) will be,

$\left[g=9.8 \mathrm{m} {\sec }^{-2}\right]$

A particle is projected from a point $(0,1)$ on $Y$ -axis (assume $+Y$ direction vertically upwards) aiming towards a point $(4, 9).$ It falls on the ground on $x$ axis in $1 \sec$. If the speed of projection is $\sqrt{\beta }\mathrm{m} {\mathrm{s}}^{-1}$ where $\beta$ is an integer. Find $\beta .$

Take $g=10 \mathrm{m} {\mathrm{s}}^{-2}$ and all coordinate in metres.

A bomber flying upward at an angle of $53°$with the vertical releases a bomb at an altitude of $800 \mathrm{m}$ The bomb strikes the ground $20 \sec$ after its release. The velocity of the bomber at the time of the release of the bomb is $v \mathrm{m} {\mathrm{s}}^{-1}$ . Find $\frac{v}{4}.$

[Given $\sin 53°=0.8; g=10 \mathrm{m} {\mathrm{s}}^{-2}$ ] 

A man is travelling on a flat car which is moving up a plane inclined at $\cos \theta =\frac{4}{5}$ to the horizontal with a speed $5 \mathrm{m} {\mathrm{s}}^{-1}$. He throws a ball towards a stationary hoop located perpendicular to the incline in such a way that the ball moves parallel to the slope of the incline while going through the centre of the hoop. The centre of the hoop is $4 \mathrm{m}$ high from the man's hand calculate the time taken (in $\mathrm{s}$) by the ball to reach the hoop in second. 

A stone is projected horizontally with speed $V$ from a height $h$ above ground. A horizontal wind is blowing in a direction opposite to the velocity of projection and gives the stone a constant horizontal acceleration $f$ (in a direction opposite to the initial velocity). As a result, the stone falls on the ground at a point vertically below the point of projection. Then find the value of $\frac{f^{2}h}{g{V}^2}$.

($g$ is acceleration due to gravity)