Find the time an aeroplane having velocity $v$ takes to fly around a square with side $a$ if the wind is blowing at a velocity $u$ along one side of the square.

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 man directly crosses a river in time $t_1$ and swims down the current at a distance equal to the width of the river in time $t_2$. If $u$ and $v$ be the speed of the current and the man, respectively, show that $\frac{t_1}{t_2}:\frac{\sqrt{v+u}}{\sqrt{v-u}}$.