Question

A man wants to reach point $B$ on the opposite bank of a 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 $B?$

• A. $u\sqrt{2}$
• B. $u/\sqrt{2}$
• C. 2 u
• D. u/2

Answer 1

Answer: (B)

Let $v$ be the speed of boatman in still water.

Resultant of $v$ and $u$ should be along $AB$. Components of $v_b$ (absolute velocity of boatman) along $x$ and $y$ directions are,
$v_x=u-v\sin \theta$
and $v_y=v\cos \theta$
Further, $\tan 45^{\circ }=\frac{v_y}{v_x}$
Or $1=\frac{v\cos \theta }{u-v\sin \theta }$
$v=\frac{u}{\sin \theta +\cos \theta }=\frac{u}{\sqrt{2}\sin \left(\theta +45^{\circ }\right)}$
$v$ is minimum at,
$\theta +45^{\circ }=90^{\circ }$ or $\theta =45^{\circ }$
and $v_{\min }=\frac{u}{\sqrt{2}}$