Question

A motor boat covers the distance between two places on a river and returns in \[14\,{\text{hrs}}\] . If the velocity of the boat in still water is \[35\,{\text{km/h}}\] and the velocity of water in river is \[5\,{\text{km/h}}\] , then the distance between the two places is:
A. \[100\,{\text{km}}\]
B. \[240\,{\text{km}}\]
C. \[220\,{\text{km}}\]
D. \[180\,{\text{km}}\]

Answer 1

Hint: First of all, we will apply the concept of relative velocity on two bodies moving in the same direction and opposite direction. We will then use the time and distance formula and substitute the values and manipulate accordingly.

Complete step by step answer:
In the given question, we are supplied the following information:
The time taken by the motor boat to move from a place and return to the same place again is \[14\,{\text{hrs}}\] .
The velocity of the boat in still water is \[35\,{\text{km/h}}\] .
The velocity of water flowing in the river is \[5\,{\text{km/h}}\] .
We are asked to calculate the distance between the two places.

For this we should understand the problem. It has both upstream and downstream journeys. Let us take the forward journey to be along the upstream and the return journey be along the downstream flow. While going upstream, the velocity of the boat decreases because the flow of water will be in the opposite direction. In going downstream, the velocity of the boat will increase as the water is also flowing in the same direction.

We know that,
\[t = \dfrac{d}{v}\] …… (1)
Where,
\[t\] indicates time taken to travel.
\[d\] distance travelled.
\[v\] indicates velocity of the boat.

Now,
Velocity of the boat in upstream motion is \[\left( {35 - 5} \right)\,{\text{m/s}}\] .
Velocity of the boat in downstream motion is \[\left( {35 + 5} \right)\,{\text{m/s}}\] .
According to the question,
Both these journeys take \[14\,{\text{hrs}}\] in total. So, we can write:
$ \dfrac{d}{{35 - 5}} + \dfrac{d}{{35 + 5}} = 14 \\ $
$ \implies \dfrac{d}{{30}} + \dfrac{d}{{40}} = 14 \\ $
$ \implies \dfrac{{7d}}{{120}} = 14 \\ $
$ \implies d = \dfrac{{14 \times 120}}{7} \\ $

On further simplification we get,
\[d = 240\,{\text{km}}\]
Hence, the distance between the two places is \[240\,{\text{km}}\] .

So, the correct answer is “Option B”.

Note:
This numerical can be solved if you have some concept of the relative velocities. It contains two parts: one in upstream and the other in the downstream. While going upstream, the boat has to put a lot of effort to keep going, so there is a decrease in its velocity. On the other hand, while going downstream, the boat flows at ease and at some extra velocity provided by the flowing water.