Question

A swimmer can swim with a speed $v$ in still water.


If the speed of the river flow is $u$ and the time taken by swimmer to reach opposite point on other bank of the river is $T_1$ and the time taken to travel an equal distance upstream (against the water current) in the river is $T_2$, then ratio $\frac{T_2}{T_1}$ will have a value

• A. $\sqrt{\frac{1-\frac{u}{v}}{1+\frac{u}{v}}}$
• B. $\sqrt{\frac{1+\frac{u}{v}}{1-\frac{u}{v}}}$
• C. $\frac{\sqrt{v^2-u^2}}{(v+u)}$
• D. $\frac{\sqrt{v^2-u^2}}{v}$

Answer 1

The correct option is B $\sqrt{\frac{1+\frac{u}{v}}{1-\frac{u}{v}}}$

Case 1- When reaching the directly opposite point:
$T_1=\frac{d}{\sqrt{v^2-u^2}}$

Case 2- When going upstream:
$T_2=\frac{d}{(v-u)}$

So, $\frac{T_2}{T_1}=\frac{\sqrt{v^2-u^2}}{v-u}=\sqrt{\frac{v^2-u^2}{(v-u{)}^2}}$
$=\sqrt{\frac{v+u}{v-u}}=\sqrt{\frac{1+\frac{u}{v}}{1-\frac{u}{v}}}$