A helicopter takes off along the vertical with an acceleration of $3 \mathrm{m} {\mathrm{s}}^{-2}$ & zero initial velocity. In a certain time, the pilot switches off the engine. At the point of take-off, the sound dies away in $30 \mathrm{s}$. Determine the velocity of the helicopter at the moment when its engine is switched off, assuming the velocity of sound is $320 \mathrm{m} {\mathrm{s}}^{-1}$.

Two swimmers start a race. One who reaches the point $C$ first on the other bank wins the race. A makes his strokes in a direction of $37°$ to the river flow with velocity $5{\mathrm{kmhr}}^{-1}$ relative to water. $B$ makes his strokes in a direction $127°$ to the river flow with same relative velocity. River is flowing with speed of $2 {\mathrm{kmhr}}^{-1}$ and is $100\mathrm{m}$ wide. Who will win the race? Compute the time taken by $A$ and $B$ to reach the point $C$ if the speed of $A$ and $B$ on the ground are $8 {\mathrm{kmhr}}^{-1}$ and $6 {\mathrm{kmhr}}^{-1}$ respectively.

A swimmer starts to swim from point $A$ to cross a river. He wants to reach point $B$ on the opposite side of the river. The line $AB$ makes an angle $60°$ with the river flow as shown. The velocity of the swimmer in still water is same as that of the water

 (i) In what direction should he try to direct his velocity? Calculate angle between his velocity and river velocity.

(ii) Find the ratio of the time taken to cross the river in this situation to the minimum time in which he can cross this river.

Two inclined planes $OA$ and $OB$ having inclinations $30°$ and $60°$ respectively, intersect each other at $O$ as in figure. A particle is projected from point $P$ with velocity $u=10\sqrt{3} {\mathrm{ms}}^{-1}$ along a direction perpendicular to plane $OA$. If the particle strikes the plane $OB$ perpendicularly at $Q$. Calculate

(i) Time of flight.

(ii) Velocity with which particle strikes the plane $OB$.

(iii) Vertical height of $P$ from $O$

(iv) Maximum height from $O$ attained by the particle.

(v) Distance $PQ$.

A projectile is launched at an angle $\alpha$ from a cliff of height $H$ above the sea level. If it falls into the sea at a distance $D$ from the base of the cliff, show that its maximum height above the sea level is 

$\left[H+\frac{D^2{\tan }^2\alpha }{4\left(H+D \tan \alpha \right)}\right]$

A particle is projected with velocity $2\sqrt{gh},$so that it just clears two walls of equal height $h$which are at a distance of $2h$from each other. Show that the time of passing between the walls is  $2\sqrt{\frac{h}{g}}$.