A man swims with avelocity `v_(mw)` in still water. When the water moves with a velocity `u_(w)(ltv_("new")^(2))` the man crosses the river to and fro in minimum time `T_(1)`. If the man intends to cross the river perpendicu larly, he takes time `T_(2)` for to and fro journey. Now he swims in the donwstream and comes back to his initial position by swimming upstream along the shore. For to and fro journey along the shore, the man takes a time `T_(3)`. find the relation between `T_(1),T_(2)` and `T_(3)` assuming equal distance overed by the man in each case.

Explanation

As derived earler, the minimum time of crossiing the river is
`t_(m)=(d)/(v_(mw))`, where , d=width of the river
Henc,e the total time for to an fro journey is
`T_(1)=2t_(m)=(2d)/(v_(mw))`
For the minimum drift, the time of crossing the river is
`t_(1)=(d)/(v_(m))` where, `v_(m)=sqrt(v_(mw)^(2)-v_(2)^(2))`
Hence, the total time required for to and fro motion is
`T_(2)=2t_(1)=(2d)/(sqrt(v_(mw)^(2)-v_(2)^(2)))`
When the man moves in downstream, his velocity is
`v_(m_(1))=v_(mw)+v_(w)` and when he moves in upstream his velocity is
`v_(m_(2))=v_(mw)-v_(w)`.
Hence, the total time for to and fro journey is
`T_(3)=r_("down")+t_("up")`.
where, `t_("down")=(d)/(v_(m_(1)))=(d)/(v_(mw)+v_(w))`
and `t_("up")=(d)/(v_(m_(2)))=(d)/(v_(mw)+v_(w))`
Hence, `T_(3)=(2dv_("mw"))/(v_("mw")^(2)-v_(2)^(2))`
Comparing the above expressions of `T_(1),T_(2)` and `T_(3)`, we have
`T_(2)^(2)=T_(1)T_(3)`