Question

A boat moves relative to water with a velocity $v$ and river is flowing with $2v$. At what angle with the stream should the boat move to have minimum drift?

• A. ${30}^{\circ }$
• B. ${60}^{\circ }$
• C. ${90}^{\circ }$
• D. ${120}^{\circ }$

Answer 1

The correct option is D ${120}^{\circ }$

Let the boat move at angle $\theta$ to the normal as shown in the figure.
Then, time to cross the river $t=\frac{l}{v\cos \theta }$
Drift $x=(2v-v\sin \theta )t=(2v-v\sin \theta )\frac{l}{v\cos \theta }$
$=l(2\sec \theta -\tan \theta )$


$\frac{dx}{d\theta }=l(2\sec \theta \tan \theta -{\sec }^2\theta )$
For minimum drift, $\frac{dx}{d\theta }=0$
$\Rightarrow 2\sec \theta \tan \theta -{\sec }^2\theta =0$
$\Rightarrow \sin \theta =\frac{1}{2}$
$\Rightarrow \theta ={30}^{\circ }$
and the angle made by boat with stream will be $\varphi ={90}^{\circ }+{30}^{\circ }={120}^{\circ }$