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A man directly crosses a river in time t...

A man directly crosses a river in time `t_1` and swims down the current a distance equal to the width of the river I time `t_2`. If `u and v` be the speed of the current and the man respectively, show that : `t_1 :t_2 = sqrt(v + u) : sqrt(v - u))`.

Text Solution

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We know that the time taken foa a man to cross the river directly is given by `t_1 = (d)/(sqrt(v^2 - u^2))`, where `d` is the width of the river.
Now, if the man swims downstream then the resultant velocity is `v + u`. The corresponding time, `t_2 = (d)/(u + v)`.
Now, `(t_1)/(t_2) = (v+u)/(sqrt(v^2 - u^2))= (sqrt(v + u) sqrt(v + u))/(sqrt(v + u)sqrt(v - u)) rArr (t_1)/(t_2) = (sqrt(v + u))/(sqrt(v - u))`.
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