A boat moves relative to water with a velocity `v` which is `n` times less than the river flow velocity `u`. At what angle to the stream direction must the boat move to minimize drifting ?

Explanation

In this problem, one thing should be carefully noted that the velocity of boat is less than the river velocity. Hence boat cannot reach the point directly opposite to its starting point i.e., drift can never be zero. Suppose boat starts at an angle `theta` from the normal direction up stream as showm.

Component of velocity of boat along the river,
`v_(x)=u-v sin theta`
and velocity perpendicular to the river,
`v_(y)=v cos theta`,
time taken to cross the river is
`t = (d)/(v_(y))=(d)/(v cos theta)`
Drift `x=(v_(x))t=(u-v sin theta)(d)/(v cos theta)`
The drift x is minimum, when `(dx)/(d theta)=0`
or `((ud)/(v))(sec. theta tan theta)-d sec^(2)theta=0`
`(u)/(v)sin theta 1 rArr sin theta =(v)/(u)`
So, for minimum drift, then must move at an angle
`theta = sin^(-1)((v)/(u))=sin^(-1)((1)/(n))`
from normal direction.