Question

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

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

Answer 1

The correct option is C $\frac{\pi }{2}+si{n}^{-1}\left(\frac{1}{n}\right)$


Let: $v_R$ = velocity of river
Velocity of boat $=v_b=\frac{v_R}{n}$
Net drift velocity $v=v_r+v_{b}cos\theta$
$=v_R+\left(\frac{v_R}{n}\right)cos\theta =v_R(1+\frac{cos\theta }{n})$
velocity along y axis $v_y=v_b\sin \theta$
$w=v_b\sin \theta .t$
$t=\frac{w}{v_{b}sin\theta }$
$d=v_R(1+\frac{cos\theta }{n})\frac{w}{v_{b}sin\theta }=w[\text{n cosec}\theta +\text{cot}\theta ]$
For minimum drift $\frac{d\left(d\right)}{dx}=0$
$\therefore [-\text{n cosec}\theta \text{cot}\theta -{\text{cosec}}^2\theta ]=0$
$\cos \theta =-\frac{1}{n};\theta =\frac{\pi }{2}+{\sin }^{-1}\left(\frac{1}{n}\right)$
So correct option is (C)