Question

A boat moves relative to water with a velocity which is $1 / n$ times the river flow velocity. At what angle to the stream direction must the boat move to minimize drifting?

• A. $\pi / 2$
• B. $\sin ^{-1}(1 / n )$
• C. $\frac{\pi}{2}+\sin ^{-1}(1 / n )$
• D. $\frac{\pi}{2}-\sin ^{-1}(1 / n )$

Answer 1

Answer: (C)

Let width of the river be d speed of stream be $v$ and the speed of the boat relative to water be $u$ and the angle with the verticle at which the boat must move for minimum drifting is $\theta$.
Time taken to cross the river $=\frac{d}{u \cos \theta}$
Drift of the boat is $( v - usin \theta)( d / u \cos \theta)$
Differentiating this w.r.t time and equating to zero we get the angle $\theta$ for minimum drifting as $\sin ^{-1}\left(\frac{ u }{ v }\right)$ Angle with the direction of the stream is $90^{\circ}+\sin ^{-1}\left(\frac{ u }{ v }\right)$
Here $u =\frac{ v }{ n }$
$\therefore $ Angle $=\frac{\pi}{2}+\sin ^{-1}\left(\frac{1}{ n }\right)$