On a river coast, there is a port; when a barge passed the port, a motor boat departed from the port to a village at the distance $S_1 = 15$ km downstream. It reached its destination after $t = 45$ mins, turned around, and started immediately moving back towards the starting point. At the distance $S_2 = 9$ km from the village, it met the barge. What is the speed of the river water, and what is the speed of the boat with respect to the water? Note that the barge did not move with respect to the water.
(Source: Jaan Kalda IPHO notes)
This is of course an easy problem and can be easily solved by writing the distance travelled by boat in upstream and downstream motion. But the solution given in the book is very different. It involves the idea of relative motion. The author wrote that “In the water’s frame of reference, it is clear that departing from the barge and returning to it took exactly the same amount of time.”
Proof of this fact: Working in water’s frame of reference, we see that the barge is at rest and the boat is travelling with uniform speed throughout the journey. Also, In the downstream and upstream motion, change in relative position will be equal and opposite. So, departing from the barge and returning to it took exactly the same amount of time.
Although I have proved this fact, I can’t understand why does the proof work. I mean to ask how can we look at this result in an intuitive way?
