Question

A river is flowing from west to east at a speed of $5m/\min$. A man on the south bank of the river, capable of swimming at $10{\text{ }}m/\min$ in still waters, wants to swim across the shortest path distance. In what direction should he swim?

Answer 1

To reach on the opposite bank of the river in the shortest direction, the swimmer should swim at right angles to the direction of the flow of the river. When a swimmer wants to get across a river in the shortest time and in the shortest path, the path is a straight line, perpendicular to the direction of the river flow. To go in that path, the swimmer must swim at an angle to compensate or offset the river currents

Complete step by step solution:
Let the width of the river be $d$. If a man swims at an angle $\theta$ with the direction of the flow of water, his velocity component is perpendicular to the direction of the flow of water that is $10sin\theta$, In order to swim across the river in the shortest direction, the man should swim straight due north. This is because the velocity of the river is west to east and there are no components in north-south. So, it will not affect the man's time in order to swim to the other bank.
Now we know that river is flowing from west to east at a speed of $5m/\min$ capable of swimming at $10m/\min$. so,
The time taken to cross the river $t = \dfrac{d}{{{v_s}\cos \theta }}$
Wants to swim across the shortest path distance that means the time to be minimum that is $\cos \theta = \max \Rightarrow \theta = 0^\circ$

Hence, the swimmer should swim due north.

Note: For the shortest time, the swimmer should swim along $AB$ so he will reach at point $C$ due to the velocity of the river. That is. He should swim due north.