Question

A man can swim a stream of width 100m in 4 minutes directly when there is no current of water and in 5 minutes where there is a current of water. The velocity of the current of water in the stream is
A. $15m{s^{ - 1}}$
B. $5m{s^{ - 1}}$
C. $2.5m{s^{ - 1}}$
D. $0.25m{s^{ - 1}}$

Answer 1

Relative velocity is defined as the vector difference between the velocities of the two bodies with respect to each other. To find the solution of the given question write down all the provided physical quantities and then carefully apply the concept of relative velocity.

Formula used:
${\vec v_{ab}} = {\vec v_a} + \vec v{}_b$

Complete step by step answer:
We will first find the expression of velocity of the man in still water then the velocity of the man in water with current. Later substituting the values obtain the velocity of the current of water in the stream.
The relative of an object is defined as the velocity of an object B in the rest frame of another object A. Mathematically, velocity of object B relative to A is given as,
${\vec v_{ab}} = {\vec v_a} + \vec v{}_b$
Velocity of the man in still water ${V_{man}} = \dfrac{{100}}{{4 \times 60}} = \dfrac{5}{{12}}m/s$
Velocity of the man in water with a current ${V_{current}} = \dfrac{{100}}{{5 \times 60}} = \dfrac{1}{3}m/s$
Now, let the velocity of the current be ‘v’
Applying the concept of relative velocity, we get
Relative velocity of man with respect to the current is given by, $\sqrt {\left[ {{v^2} + {{\left( {\dfrac{5}{{12}}} \right)}^2}} \right]}$
But, the velocity of this man with respect to the current is calculated as $\dfrac{1}{3}m/s$
So, we get the equation $\Rightarrow \sqrt {\left[ {{v^2} + {{\left( {\dfrac{5}{{12}}} \right)}^2}} \right]} = \left( {\dfrac{1}{3}} \right)$
On solving this equation, we get the velocity of the current of water in the stream $v = \dfrac{1}{4} = 0.25m/s$

So, the correct answer is “Option D”.

Note:
All the motions are relative to some frame of reference. For example, if a body is at rest it means that it is not in motion. This means that it is being described with respect to a frame of reference which is moving together with the body.