At $t=0$, two particles $A$ and $B$ start moving with same speed along the sides of an equilateral triangle as shown in figure. Find the velocity of approach of $A$
w.r.t. $B$ at time

(i) $t=0$ and

(ii) $t=5 \sec$

A rocket moves horizontally with a constant velocity $u$ at a height $h$. A guided missile is fired vertically with a speed when the rocket passes above it.

Assuming that the missile always aims at the rocket with the constant speed $v$, find the time after the missile strikes the rocket.

A $400 \mathrm{m}$ wide river is flowing at a rate of $2.0 \mathrm{m} {\mathrm{s}}^{-1}$. A boat is sailing with a velocity of $10 \mathrm{m} {\mathrm{s}}^{-1}$ with respect to the water, in a direction perpendicular to the river.

(a) Find the time taken by the boat to reach the opposite bank.

(b) How far from the point directly opposite to the starting point does the boat reach the opposite bank?

(c) In what direction does the boat actually move?

A man wishing to cross a river flowing with velocity $u$ jumps at an angle $\theta$ with the river flow.

(a) Find the net velocity of the man with respect to the ground if he can swim with speed $v$ in still water.

(b) In what direction does the man actually move?

(c) Find how far from the point directly opposite to the starting point does the boat reach the opposite bank, if the width of the river is $d.$

To a man running upwards on the hill, the rain appears to fall vertically downwards with $4 \mathrm{m} {\mathrm{s}}^{-1}$. The velocity vector of the man with respect to earth is $\left(2\hat{\mathrm{i}}+3\hat{\mathrm{j}}\right) \mathrm{m} {\mathrm{s}}^{-1}$ if the man starts running down the hill at the same speed, then determine the relative speed of the rain with respect to man.

An aeroplane flies along a straight path $A$ to $B$ returns back again. The distance between $A$ and $B$ is $l$ and the aeroplane maintains the constant speed $v$ with respect to the wind. There is a steady wind with a speed $u$ at an angle $\theta$ with line $AB$. Determine the expression for the total time of the trip.

A pilot is supposed to fly a certain distance $AB$, due east from $A$ to $B$ and then due west from $B$ to $A$. The velocity of plane is $v$ and that of air is $u$. The time for the round trip is $t_0$ in still air. Show that

(a) If the air velocity be due east or west, the time for the round trip will be $t_1=\frac{t_0}{\left(1-{\displaystyle \frac{u^2}{v^2}}\right)}$

(b) if the air velocity is due north or south, the time for a round trip will be $t_2=\frac{t_0}{\sqrt{\left(1-\frac{u^2}{v^2}\right)}}.$