Question

A streamer goes downstream and covers the distance between two parts in 5 hours, while it covers the same distance upstream in 7 hours. If the speed of the stream is 3 km/h, then the speed of the streamer in still water is?

Answer 1

Hint: Assume speed of streamer in still water $=u$km/h and speed of stream $=v$ km/h. Then calculate the speed of the streamer while moving upstream and downstream. Use \[\text{time taken = }\dfrac{\text{distance travelled}}{\text{speed}}\] to calculate speed.

Complete step-by-step answer:
The main concept behind this type of question is that when an object moves in the direction of water current (downstream) then the net speed of object is the sum of its speed in still water and that of stream and when it flows in the direction opposite to that of water current (upstream) then the net speed of object is the difference of speed of object and that of stream. This concept is known as relative velocity.

Let speed of streamer in still water be ‘\[u\]’ km/h,
 Speed of stream be ‘\[v\]’ km/h
And, the distance travelled by the streamer be ‘$d$’ km.
Therefore, speed of streamer travelling upstream = \[\left( u-v \right)\]km/h
And, speed of streamer travelling downstream = \[\left( u+v \right)\]km/h
We know that; \[\text{distance travelled = speed}\times \text{time taken}\]
Hence, \[\text{time taken = }\dfrac{\text{distance travelled}}{\text{speed}}.\]
Considering the journey of streamer downstream,
\[\text{time taken = }\dfrac{\text{distance travelled}}{\text{speed}}.\]
$\begin{align}
  & \therefore 5=\dfrac{d}{u+v} \\
 & d=5(u+v)..................(i) \\
\end{align}$
Now, considering the journey of streamer upstream,
\[\text{time taken = }\dfrac{\text{distance travelled}}{\text{speed}}.\]
$\begin{align}
  & \therefore 7=\dfrac{d}{u-v} \\
 & d=7(u-v)................(ii) \\
\end{align}$
From equation (i) and (ii), we get,
$\begin{align}
  & 5(u+v)=7(u-v) \\
 & 5u+5v=7u-7v \\
 & 2u=12v \\
 & u=6v \\
\end{align}$
We have been provided, $v=3\,\text{ km/h}$. Substituting this value in the above equation, we get,
$u=6\times 3=18\text{ km/h}$
Hence, the speed of the streamer in still water is 18 km/h.

Note: It is important to note that when any object travels upstream its speed decreases and the time of journey increases because the flow of stream opposes the motion of object and when the object travels downstream its speed increases and the time of journey decreases because the flow of stream favors the motion of object.