A man crosses the river, perpendicular to river flow in time $t$ seconds and travels an equal distance down the stream in $T$ second. The ratio of man's speed in still water to the speed of river water will be:

\frac{t^{2} - T^{2}}{t^{2} + T^{2}}

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\frac{T^{2} - t^{2}}{T^{2} + t^{2}}

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\frac{t^{2} + T^{2}}{t^{2} - T^{2}}

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\frac{T^{2} + t^{2}}{T^{2} - t^{2}}

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Solution

The correct option is C $\frac{t^{2} + T^{2}}{t^{2} - T^{2}}$
Let velocity of man in still water be $v$ and that of water with respect to ground be $u$ .

Velocity of man perpendicular to river flow with respect to ground $= \sqrt{v^{2} - u^{2}}$

Velocity of man downstream $= v + u$

As given, $= \sqrt{v^{2} - u^{2}} t = (v + u) T$

$\Rightarrow (v^{2} - u^{2}) t^{2} = {(v + u)}^{2} T^{2} \Rightarrow (v - u) t^{2} = (v + u) T^{2}$

$∴ \frac{v}{u} = \frac{t^{2} + T^{2}}{t^{2} - T^{2}}$

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