Question Number 14920 by Tinkutara last updated on 05/Jun/17

A swimmer crosses a flowing river of  width d to and fro in time t_1 . The time  taken to cover the same distance up  and down the stream is t_2 . If t_3  is the  time the swimmer would take to swim  a distance 2d in still water, then prove  that t_1 ^2  = t_2 t_3 .

Answered by ajfour last updated on 05/Jun/17

Commented byajfour last updated on 05/Jun/17

to go to the other bank and return  back to same point (to and fro)  swimmer must swim at an angle  θ (see fig.) such that     vsin 𝛉=u     ...(i)    t_1 =((2d)/(vcos 𝛉))      ....(ii)  to go upstream a distance d and  return  back time taken is     t_2 =(d/(v−u))+(d/(v+u))    ....(iii)  to swim a distance 2d in still  water time taken is    t_3 =((2d)/v) .    t_2 t_3 =((2d^2 )/v)(((2v)/(v^2 −u^2 ))) =((4d^2 )/(v^2 −v^2 sin ^2 θ))          =  (((2d)/(vcos θ)))^2  = t_1 ^2                         [see (ii), (iii), and (i) ].

Commented byTinkutara last updated on 05/Jun/17

Thanks Sir!