A swimmer wishes to cross a $500 \mathrm{m}$ wide river flowing at $5 \mathrm{km} {\mathrm{h}}^{-1}$. His speed with respect to water is $3 \mathrm{km} {\mathrm{h}}^{-1}$. Suppose he heads in a direction making an angle $\text{θ}$ with the flow. The man has to reach the other shore at the point directly opposite to his starting point. If he reaches the other shore somewhere else, he has to walk down to this point. Find the minimum distance that he has to walk.

An aeroplane has to go from a point $A$ to another point $B$, $500 \mathrm{km}$ away due $30°$ east of north. The wind is blowing due north at a speed of $20 \mathrm{m} {\mathrm{s}}^{-1}$. The air-speed of the plane is $150 \mathrm{m} {\mathrm{s}}^{-1}$.

(a) Find the direction in which the pilot should head the plane to reach the point $B$.

(b) Find the time taken by the plane to go from $A$ to $B$.

Previous Problem: Two friends $A$ and $B$ are standing at a distance $x$ apart in an open field and the wind is blowing from $A$ to $B$. $A$ beats a drum and $B$ hears the sound at time $t_1$ after he sees the event. $A$ and $B$ interchange their positions and the experiment is repeated. This time, $B$ hears the drum at time $t_2$ after he sees the event. Calculate the velocity of sound in still air $v$ and the velocity of wind $u$. Neglect the time that light takes in travelling between the friends.

Consider $A$ and $B$ in the previous problem. Change their positions in such a way that the line joining them becomes perpendicular to the direction of the wind while maintaining the separation $x$. What will be the time lag $B$ finds between seeing and hearing the drum beating by $A$?

Six particles, situated on the corners of a regular hexagon of side $a$, move at a constant speed $v$. Each particle maintains a direction towards the particle at the next corner. Calculate the time the particles will take to meet each other.