Question

A man can swim with velocity v relative to water.He has to cross a river of width d flowing with a velocity $u(u>v)$. The distance through which he is carried down stream by the river is x.Which of the following statement is correct.

• A. If he crosses the river in minimum time $x=\frac{du}{v}$
• B. x can not be less than $\frac{du}{v}$
• C. For x to be minimum he has to swim in a direction making an angle of $\frac{\pi }{2}+si{n}^{-1}\left(\frac{v}{u}\right)$ with the direction of the flow of water
• D. x will be max, if he swims in a direction making an angle of $\frac{\pi }{2}+si{n}^{-1}\frac{v}{u}$ with direction of the flow of water

Answer 1

Answer: (A), (C)

The correct options are
A If he crosses the river in minimum time $x=\frac{du}{v}$
C For x to be minimum he has to swim in a direction making an angle of $\frac{\pi }{2}+si{n}^{-1}\left(\frac{v}{u}\right)$ with the direction of the flow of water

The time required for crossing = $t=\frac{d}{v}cos\theta$.

$x=(u-vsin\theta )t=u\frac{d}{v}sec\theta -dtan\theta$.

For x to be minimum,$\frac{dx}{d\theta }=0$.

This gives $sin\theta =\frac{v}{u}$

ie,

angle needs to be =$\frac{\pi }{2}+si{n}^{-1}\left(\frac{v}{u}\right)$

For minimum time crossing $x=\frac{du}{v}$