A river is flowing from west to east at a speed of $5$ metres per minute. A man on the south bank of the river, capable of swimming at $10$ metres per minute in still water, wants to swim across the river in the shortest time. He should swim in a direction

Step 1: Define the variables and parameters.

Let the width of the river be $W$, the speed of the river flow be $v_r$, and the swimming speed of the man in still water be $v_s$.

$$v_r = 5\text{ m/min}$$ $$v_s = 10\text{ m/min}$$

Step 2: Express the time taken to cross the river.

Let $\theta$ be the angle that the swimming direction makes with the direction perpendicular to the river flow (due North).

$$v_y = v_s \cos\theta$$ $$t = \frac{W}{v_s \cos\theta}$$

Step 3: Determine the condition for minimum time.

For the crossing time $t$ to be minimum, the term $\cos\theta$ must be maximized.

$$\cos\theta = 1$$ $$\theta = 0^\circ$$

This means the man must swim directly perpendicular to the river flow.

Answer: Due North