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 case, the velocity of boat is less than the river flow velocity. Hence, boat cannot reach the point directly opposite to its starting point, i.e., drift can never be zero.
Suppose the boat starts at an angle `theta` from the normal diraction up stream as shown in (Fig. 5.90).
Component of the 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) = (ud)/(v) sec theta - d tan theta`
The drift `x` is minimum when `(dx)/(d theta) = 0`
or `((ud)/(v)) (sec theta . tan theta) - d sec^2 theta = 0`
or `(u)/(v) sin theta = 1 rArr sin theta = (v)/(u)`
i.,e., for minimum drift, the boat must move at an angle `theta = sin^-1 theta`
=`sin^-1 ((v)/(u)) = sin^-1 (1)/(n)` from the normal direction.
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