Question

A river is flowing from west to east at a speed of $5\text{m/min}$. A man on the south bank of the river, capable of swimming at $10\text{m/min}$ in still water, wants to swim across the river in the shortest time. He should swim in a direction

• A. due north
• B. ${30}^{\circ }$ east of north
• C. ${30}^{\circ }$ west of north
• D. ${60}^{\circ }$ east of north

Answer 1

Answer: (A)

Step 1: Given that:

Velocity of the river($v_{river}$) from west to east = $5mmi{n}^{-1}$

Velocity of the person from the south bank of the river in still water($v_{man}$) = $10mmi{n}^{-1}$

Step 2: Finding the direction for the man to swim across the river in the shortest time:

The shortest distance takes the shortest time for the person in the river.

The situation can be drawn as follows;

Here, AC is the shortest distance for the person to cross the river from the south bank of the river.

The component of velocity of the person along the shortest path = $v_{man}\cos \theta$

Now, the time taken by the person to cross the river along shortest path is given as;

$t=\frac{AC}{v_{man}\cos \theta }$

The time will be minimum when the denominator will be maximum.

That is basically, $\cos \theta$ = maximum

Maximum value of $\cos \theta$ = 1

Therefore

$\cos \theta =1$

$\cos \theta =\cos 0^0$

$\theta =0^0$

That is in the north direction.

Thus,

The direction in which the swimmer should move will be in north direction.

Thus,

Option a) due north is the correct option.