# A swimmer crosses a flowing stream of width 'w' to and fro in time $t_1$. The time taken to cross the same distance up and down the stream in $t_2.$ If $t_3$ is the time the swimmer would take to swim a distance $2w$ in still water. Then

$(a)\;{t_1}^2 =t_2t_3\quad (b)\;{t_2}^2=t_1t_3 \quad (c)\;{t_3}^2=t_1t_2 \quad (d)\;t_3=t_1+t_2$

Let 'u' be the velocity of swimmer in still water and 'v' be velocity of river.
In case I
Since v and u are $\perp$ to each other.
$t_1=2 \bigg(\large\frac{w}{\sqrt {u^2-v^2}}\bigg)$
In case II
the swimmer goes upstream and downstream
$t_2=\large\frac{w}{u+v}+\frac{w}{u-v}$
$\qquad=\large\frac{2uw}{u^2-v^2}$
In case III
Still water
$t_3=\large\frac{2w}{u}$
We see ${t_1}^2=t_2t_3$
Hence a is the correct answer.

edited Jan 25, 2014 by meena.p