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))`.

Explanation

A man will row directly across a flowing river if his resultant velocity or river flow and the manis along ` OC`, which is perpendicular to the river velcity u. Fig. 2 (c ) .68.
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Resultant velcity of man across the river
`=sqrt u^(2) -u^(2)`
Resultant velcity of man down the stram
` =u +v`
If (S) is thedistance coverd in each case, then
`t_(1) =S/(sqrt(u^(2) -v^(2)) =and t_(2) = S/(u+v)`
`t_(1)/t_(2) = (u+v)/(sqrt(u^(2) -v_(2)) =(u+v)/(sqrt( (u-v) (u+v)) =(sqrt u+v)/sqrt(u-v)`
:. t_(1) : t_(2) =sqrt (u +_v) : sqrt (u-v)`.