A man wants to reach point B on the opposite bank of a river flowing at a speed as shown in figure. What minimum speed relative to water should the man have so that he can reach point B? In which direction should he swim?

Explanation

Let v be the speed of boatman in still water.

Resultant of v and u should be along AB. Components of `v_(b)` (absolute velocity of boatman) along x and y direction are,
`v_(x)=u-v sin theta` and `v_(y)=v cos theta`
Further, `tan 45^(@)=(v_(y))/(v_(x))`
or `1=(v cos theta)/(u-v sin theta)`
`therefore v=(u)/(sin theta+ cos theta)=(u)/(sqrt(2)sin(theta+45^(@)))`
v is minimum at,
`theta+45^(@)=90^(@)` or `theta = 45^(@)` and `v_(min) = (u)/(sqrt(2))`