question_answer

A swimmer crosses a river of width d flowing at velocity V. While swimming, he keeps himself always at an angle of \[{{120}^{o}}\] with the river flow and on reaching the other end he finds drift of d/2 in the direction of flow of river. The speed of the swimmer with respect to the river is:

A)  \[(2-\sqrt{3})\,V\]       

B)  \[2(2-\sqrt{3})\,V\]

C)  \[4\,(2-\sqrt{3})\,V\]               

D)  \[(2+\sqrt{3})\,V\]

Correct Answer: C

Solution :

Drift = d/2             \[\frac{d}{2}=({{V}_{r}}-{{V}_{s}}\,\sin \,{{30}^{o}})\,d/Vs\,\cos \,{{30}^{o}}\]             \[\frac{d}{2}=\left( V-\frac{{{V}_{s}}}{2} \right)\frac{d}{{{V}_{s}}\sqrt{3/2}}\]             \[\frac{d}{2}=\left[ \frac{2V-{{V}_{s}}}{2}\,\frac{2d}{\sqrt{3}{{V}_{s}}} \right]\]             \[\sqrt{3}{{V}_{s}}=9V-2{{V}_{s}}\]             \[\left( \sqrt{3}+2 \right){{V}_{s}}=4V\] \[{{V}_{s}}=\frac{4}{\sqrt{3}+2}V=\frac{4(2-\sqrt{3})}{\left( \sqrt{3}+2 \right)\,\left( 2-\sqrt{3} \right)}V\]\[\Rightarrow \]        \[{{V}_{S}}=4\,(2-\sqrt{3})V\] Here =VmG = Velocity of swimmer respect to ground VrG = velocity of river with respect to ground and Vmr= velocity of swimmer with respect to river.2