Question

A swimmer capable of swimming with velocity v relative to water jumps in a flowing river having velocity u. The man swims a distance d downstream and returns back to the original position. Find out the time taken in complete motion.
A. $\dfrac{2dv}{{{v}^{2}}-{{u}^{2}}}$
B. $\dfrac{dv}{{{v}^{2}}-{{u}^{2}}}$
C. $\dfrac{3dv}{{{v}^{2}}-{{u}^{2}}}$
D. $\dfrac{4dv}{{{v}^{2}}-{{u}^{2}}}$

Answer 1

We are given the case of a relative motion. Find the relative velocity of the swimmer with respect to the river’s velocity separately for downstream and upstream motion. Recall that the distance covered divided by velocity will give you time and thus find the time taken for upstream and downstream motion separately. Now you could simply add them to get the required answer.

Formula used:
$v=\dfrac{d}{t}$

Complete answer:
We are given that a swimmer who jumps in a river that is flowing with velocity u. Also, this swimmer is capable of swimming with velocity v relative to water. It is also said that the swimmer swims distance d downstream and returns back upstream swimming the same distance d to the original position. We are asked to find the total time taken in completing this motion.
Recall that relative motion is the motion of one body with respect to another body that is either stationary or in motion. This basically depends on the frame of reference. Let us consider the case of a passenger sitting in a moving train. This passenger is at zero velocity relative to the airplane but relative to a stationary observer at the station, he is moving with same velocity as the train with respect to the ground. Similarly, in this question, we are discussing the relative motion between the river and the swimmer.
Let us consider the motion upstream and downstream the flowing river separately.
For swimming downstream,

The velocity of the swimmer with respect to the river is given by,
${{v}_{down}}=v+u$
We know that,
Distance =velocity $\times$ time
Time taken to travel distance d downstream is,
${{t}_{down}}=\dfrac{d}{v+u}$ ……………………….. (1)
For swimming upstream,

The velocity of the swimmer with respect to the river is given by,
${{v}_{up}}=v-u$
Time taken to travel distance d upstream is,
${{t}_{up}}=\dfrac{d}{v-u}$ ………………………… (2)
Adding (1) and (2) we get the total time taken for motion upstream as well as downstream as,
$T={{t}_{down}}+{{t}_{up}}=\dfrac{d}{v+u}+\dfrac{d}{v-u}$
$T=d\left( \dfrac{1}{v+u}+\dfrac{1}{v-u} \right)=d\left( \dfrac{v-u+v+u}{{{v}^{2}}-{{u}^{2}}} \right)$
$T=\dfrac{2dv}{{{v}^{2}}-{{u}^{2}}}$
Therefore, the time taken to complete motion in the given question is,
$T=\dfrac{2dv}{{{v}^{2}}-{{u}^{2}}}$

Hence, the answer to the given question is option A.

Note:
You should note that the velocity of the man is given relative to that of the river. In such a condition, one should find the answer keeping in mind that he is dealing with relative motion. All you have to take care of is that you take the relative velocity of the body while substituting for velocity.