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

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 $=t=\frac{d}{V_y}=\frac{500}{V\cos \theta }$
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, $\theta$ should be equal to $0^{\circ }$.
Hence the correct option is (a)