Question

A boat moves relative to water with a velocity $u$ and river is flowing with $2u$. 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 ${120}^{\circ }$

The given condition can be shown as


Let the boat move at an angle $\theta$ with the normal to the river flow with the velocity $um/s$ as shown in the figure.
Then, time to cross the river is $t=\frac{l}{u\cos \theta }$
and net velocity of the boat in the direction of river flow is
$2u-u\sin \theta$
$\therefore$ drift of the boat is given by
$x=(2u-u\sin \theta )\left(\frac{l}{u\cos \theta }\right)$
For $x$ to be minimum,
$\frac{dx}{d\theta }=0$
$\Rightarrow l(2\sec \theta \tan \theta -{\sec }^2\theta )=0$
$\Rightarrow \sin \theta =\frac{1}{2}$
$\Rightarrow \theta ={30}^{\circ }$ and $\varphi =90+30={120}^{\circ }$ with the stream