Question

A swimmer wishes to cross a river that is $500m$ wide flowing at a rate $u$. For this, he makes an angle $\theta$ with the vertical to the direction of flow of the river as shown. His speed with respect to still water is $v$. In order to cross the river in minimum time, the value of $\theta$ in degrees should be

• A. $0^{\circ }$
• B. ${45}^{\circ }$
• C. ${60}^{\circ }$
• D. ${90}^{\circ }$

Answer 1

The correct option is A $0^{\circ }$



The swimmer swims with a velocity $v$ making an angle of $\theta$ with the vertical.

Hence, the $Y$ component of the velocity $=v_y=v\cos \theta$

The distance along the $Y$ direction is the width of the river $=d=500m$

The time taken to cross the river $=t=\frac{d}{v_y}=\frac{500}{v\cos \theta }$

So $t\propto \frac{1}{cos\theta }$

When $\theta =0^{\circ },\cos \theta$ is maximum, hence $t$ is minimum. As $\theta$ increases, the value of $\cos \theta$ decreases, which in turn increases the value of $t$.

Hence, for time taken $t$ to be minimum, $cos\theta$ should be maximum, $\theta$ should be equal to $0^{\circ }$.

Hence the correct option is $\left(a\right)$