Irodov Problem 1.1

There are two ways of solving this problem, i) from the point of view of an observer on the raft ii) from the point of view of a person on the ground. I will provide both these methods since, I have got a lot of questions on this problem from the readers.

Since the raft is floating on the river, it moves at the same velocity as the river. Let the velocity of the river be s. Let the velocity of the boat in still water be v. This means that the boat's downstream velocity as seen from the ground will be v+s and it upstream velocity as seen from the ground will be v-s.

i) Solution From Reference Frame on the Raft

As seen by an observer on the raft, when the boat moves upstream its velocity will be v (v+s-s)  and it will be -v (-v+s-s) when its moving downstream. Here, positive sign means that the boat is moving away from the raft and negative sign means that the boat is moving towards the raft. 

Figure 1 shows the Time-Displacement diagram of the boat and the raft as seen from the point of view of the raft. The slopes of the both upstream and downstream paths of the boat relative to the raft are v and -v.  Let t1 be the time takes by the raft to meet the boat after it turns around. As seen from the figure, it is obvious that



Thus, thus the total time between the two meetings of the raft and the boat is

Now the total distance traveled by the raft is l. This means that,

ii) Solution From Reference Frame on the Ground
The Time-Displacement diagram as seen from the ground of the raft and the boat are shown in Figure 2.

The raft moves at a constant velocity of s downstream (the slope of the line is s) and hence its path is represented by a straight blue line in Figure 2. The boat moves at a speed v+s when  going upstream, represented by a line of slope v+s in Figure 2. The boat turns around and moves at a velocity -v+s when going downstream, represented by a line of negative slope of v-s.

From Figure 2 we can see that,

Now the total distance traveled by the raft is l. This means that,