Question

A man wants to reach point B on the opposite bank of river flowing at a speed as shown in figure. What minimum speed relative to water should the man have so that he can reach point B?

Answer 1

Let v be the speed of boatman in still water.
Resultant of $v$ and $u$
should be along AB. Components of $\overrightarrow{v_b}$
(absolute velocity of boatman ) along x and y-direction are,
$v_x=u-v\sin \theta$
and $v_y=v\cos \theta$
Further, $\tan {45}^{\circ }=\frac{v_y}{v_x}$
or $1=\frac{v\cos \theta }{u-v\sin \theta }$
$\therefore v=\frac{u}{\sin \theta +\cos \theta }$
$=\frac{u}{\sqrt{2}\sin (\theta +{45}^{\circ })}$
$v$ is minimum at, $\theta +{45}^{\circ }={90}^{\circ }$
or $\theta +{45}^{\circ }$
and $v_{min}=\frac{u}{\sqrt{2}}$