Question

A steamer moves with velocity $3\,km/h$ in and against the direction of river water whose velocity is $2\,km/h$ . Calculate the total time for the total journey if the boat travels $2\,km$ in the direction of a stream and then back to its place:
(A) $2\,hrs$
(B) $2.5\,hrs$
(C) $2.4\,hrs$
(D) $3\,hrs$

Answer 1

Hint Basically, the question has given us the value of velocity of the steamer as $3\,km/h$ and the velocity of the water is given to be $2\,km/h$ . To find the time taken by the steamer to complete the journey, take its velocity while going with the flow as the sum of both the velocities and find the time taken to cover a distance of $2\,km$ and then, for the roundtrip journey, take the velocity as the difference of the two given velocities and find the time taken to travel $2\,km$ with the new velocity against the flow of the river.

Complete step by step answer
As explained in the hint section of the solution to the asked question, we will start by first defining the two velocities of the steamer, one when going with the flow, which will be the addition of the two given velocities, while the other one will be when going against the flow, which will be the difference of the two velocities and then we will use these two velocities to find out the time taken to complete the whole mentioned journey.
As given in the question, the velocity of steamer is, ${v_s} = 3\,km/h$
The velocity of river’s flow is given to be, ${v_r} = 2\,km/h$
The velocity of the steamer when going with the flow can be found out as:
$
  {v_w} = {v_s} + {v_r} \\
   \Rightarrow {v_w} = 3 + 2\,km/h \\
   \Rightarrow {v_w} = 5\,km/h \\
 $
Now that we have defined the velocity of steamer with the flow, let us define the velocity of the steamer when it is going against the flow of the river as:
$
  {v_a} = {v_s} - {v_r} \\
   \Rightarrow {v_a} = 3 - 2\,km/h \\
   \Rightarrow {v_a} = 1\,km/h \\
 $
Now, if we have a look at the journey, the distance of the other point is given to be $2\,km$ from the initial position of the steamer.
Let us assume that the first part of the journey will be with the flow of the river.
To find the time taken in this part of the journey, we need to use the value of velocity of the steamer as the velocity of the steamer with the flow of the river, ${v_w}$ . Mathematically, this can be represented as:
${t_1} = \dfrac{d}{{{v_w}}}$
Where, $d$ is the distance which is given to be $d = 2\,km$ in the question.
Substituting the already found out value of ${v_w}$ , we get:
${t_1} = \dfrac{2}{5} = 0.4\,hrs$
Now that we have found out the time taken for the first part of the journey, let’s do the same for the second part.
Now the journey will be against the flow of the river, hence the velocity will be the one as against the flow of the river, ${v_a}$ while the distance will be the same as the one in the first part of the journey, $d = 2\,km$
Now, let’s find out the time taken in this part of the journey:
${t_2} = \dfrac{d}{{{v_a}}}$
Substituting the values of distance and velocity, we get:
${t_2} = \dfrac{2}{1} = 2\,hrs$
Hence, the total time of the journey becomes:
$
  t = {t_1} + {t_2} \\
   \Rightarrow t = 0.4 + 2\,hrs \\
   \Rightarrow t = 2.4\,hrs \\
 $

As we can see, the value of time taken matches with the one given in option (C). Hence, option (C) is the correct answer to the question.

Note Many students will get confused into substituting the value of the distance in the second part of the journey believing that the water at those two points is somewhere else now. While that is true, the two positions are actually relative to the ground under the water of the river and not relative to the river's water, hence, distance in both the parts of the journey will be the same.