A man can swim directly across a stream of width S met resin t seconds when there us current and time t' when there is current.show that the velocity of the current is S×√{1/t²-1/t'²}?

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Answer 1 · 2018-06-20T12:38:21

Please see the proof below

The speed of the swimmer is

#u=s/t#

#t=s/u#

Let the speed of the current be #=v#

The distance travelled by the current is

#d=vt'#

The distance travelled by the swimmer when there is a current is

#d=sqrt(s^2+d^2)=sqrt(s^2+v^2t'^2)#

The new time is

#sqrt(s^2+v^2t'^2)=ut'#

Therefore,

Squaring both sides

#s^2+v^2t'^2=u^2t'^2#

#u^2t^2+v^2t'^2=u^2t'^2#

#v^2t'^2=u^2(t'^2-t^2)=s^2/t^2(t'^2-t^2)#

#v^2=1/(t'^2t^2)(t'^2-t^2)s^2#

#v^2=(1/t^2-1/(t'^2))s^2#

#v=s sqrt(1/t^2-1/(t'^2))#

#QED#