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A battalion of soldiers is ordered to swim across a river `500 m` wide. At what minimum rate should they swim perpendicular to river flow in order to avoid being washed away by the waterfall 300 m downstream. The speed of current being \[3{\text{m/sec}}\]:
A) \[6\,{\text{m/sec}}\]
B) \[5\,{\text{m/sec}}\]
C) \[{\text{4}}\,{\text{m/sec}}\]
D) \[2\,{\text{m/sec}}\]

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Last updated date: 13th Jun 2024
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Answer
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Hint: In this solution, we will first calculate the time required by the soldiers to be displaced 300 metres downstream. The speed of the soldiers must be such that they can cross the river in this amount of time.
Formula used: In this solution, we will use the following formula:
$v = \dfrac{d}{t}$ where $v$ is the velocity of an object, $d$ is the distance it travels, and $t$ is the time.

Complete step by step answer:
We’ve been given that a battalion of soldiers is ordered to swim across a river `500 m` wide perpendicular to it such that they can avoid being washed by the current to the waterfall that is 300 m downstream.
So, the soldiers should have such a minimum velocity such that they can cross the time that they take to cross the river before they get washed away would be
$t = \dfrac{d}{v}$
$ \Rightarrow t = \dfrac{{300}}{3} = 100\,s$
So, they would be washed towards the waterfall in 100 seconds by the flow of water in the stream. Hence, they should have a velocity such that they can cross in the river (500 m) in 100 seconds which can be calculated as
$v = \dfrac{{500}}{{100}}$
$ \Rightarrow v = 5\,{\text{m/s}}$

 Hence they need to have a velocity of \[5\,{\text{m/sec}}\] so the correct choice is option (B).

 Note: In such questions, it is always helpful to realize that the motion of the soldiers moving perpendicular to the stream of the river will have independent parallel and perpendicular motion (with respect to the stream direction). The parallel direction velocity will wash them downwards towards the stream while the perpendicular direction velocity will help them in crossing the stream.