CONCEPT APPLICATION EXERCISE 5.5

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 $d.$

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 $\frac{t_1}{t_2}:\frac{\sqrt{v+u}}{\sqrt{v-u}}$.

A person rows a boat across a river making an angle of $60°$ with the downstream. Find the percentage time he would have saved, had he crossed the river in the shortest possible time.

A person, intending to cross a river by the shortest path, starts at an angle $\alpha$ with the downstream. If the speed of the person is less than that of the water current, show that $\alpha$ must be obtuse.