Question

A swimmer can swim in still water with speed $v$ and the river flowing with velocity $\frac{v}{2}$. To cross the river in shortest distance, he should swim making an angle $\theta$ with the upstream. What is the ratio of the time taken to swim across in the shortest time to that in swimming across over shortest distance ?

• A. $\sin \theta$
• B. $\cos \theta$
• C. $\tan \theta$
• D. $\cot \theta$

Answer 1

The correct option is A $\sin \theta$

Given that,
Speed of swimmer in still water, $v_b=v$
Speed of river water, $v_r=\frac{v}{2}$
Angle of velocity of swimmer with upstream $=\theta$
Let the width of the river is $W$.

The swimmer will cross the river by covering the shortest distance if the velocity of swimmer with respect to the water is perpendicular to the upstream.
The relative velocity of the swimmer with respect to water is
$v_b\sin \theta$
Time taken to cross the river in the shortest distance, $t=\frac{W}{v_b\sin \theta }$

Shortest time taken to cross the river is
$t^{\prime }=\frac{W}{v_b}$
Ratio of times taken for shortest time to that of shortest path
$\frac{t^{\prime }}{t}=\frac{\frac{W}{v_b}}{\frac{W}{v_b\sin \theta }}=\sin \theta$

Hence, option (a) is correct.