Question

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

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

Answer 1

The correct option is A $\sin \theta$

The correct option is A.

$Letdistance=d$

The angle made with the upstream=$\theta$

Therefore,

$90-\theta$=Angle made with the normal stream

Hence,

Shortest Time Method:

Time taken, t=$\frac{d}{v}$

Shortest distance method:

Time taken = t'=$\frac{d}{vcos(90-\theta )}$

=$\frac{d}{vsin\theta }$

Therefore, required ratio

= $\frac{t}{t^{\prime }}$

=$\frac{\frac{d}{v}}{\frac{d}{v}sin\theta }$

=$sin\theta$