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 gt v).` The distance traveled by man along the flow is x, when he reaches to opposite bank. For `x` to be minimum the person should swim at an angle `alpha` with the direction of the flow of water, where `alpha` is

Options

A. `sin^(-1)(v//u)`
B. `sin^(-1)(u//v)`
C. `(pi)/2+sin^(-1)(v//u)`
D. `(pi)/2+sin^(-1)(u//v)`

Answer

C

Explanation

`y:d=vcos theta timplies t=d/(v cos theta)`
`x:x=(u-vsin theta)t=(u-v sin theta)d/(vcos theta)`
`=(u/vsec theta-tan theta) d`
For `x` to be minimum
`(dx)/(d theta)=d(u/v sec theta tan theta-sec^(2) theta)=0`
`u/v tan theta=sec theta`
`sin theta=v/u`
`alpha=90^(@)+ theta=(pi)/2+sin^(-1)(v/u)`