River problems

A boy wants to swim across a river to the other bank. If his goal is to minimise the time taken to cross, it would take him 60 60 minutes. If his goal is to minimise his displacement from the starting point, it would take him 180 180 minutes.

If the width of the river is 3 2 k m 3√2km , find the speed of the river current (in kmph).

Note: •There is a steady river current perpendicular to the width of the river.

•In both these attempts, the boy swims with the same speed relative to the river current.


The answer is 4.

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1 solution

Parth Sankhe
Nov 23, 2018

Here v v = Boy's speed and u u is the current speed.

To minimise time, he just swims perpendicular to the current

3 2 v = 1 h r \frac {3√2}{v}=1hr - (1)

v = 3 2 k m p h v=3√2kmph

Notice that in the above case, since he is swimming straight, the current will drive him downstream.

To minimise the displacement, he has to swim at such an angle to the current ( θ ) \theta ) such that his velocity along the current cancels out the river current, which will not make him go downstream, thus minimising his overall displacement.

Thus, v cos θ = u v\cos \theta =u - (2)

And, 3 2 v sin θ = 3 h r s \frac {3√2}{v\sin \theta }=3hrs - (3)

sin θ = \sin \theta = ⅓

Therefore, cos θ = 8 9 \cos \theta = \frac {√8}{9}

Putting the value of v , θ v, \theta in (2), we get u = 4 k m p h u=4kmph

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