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Relative motion

The River and Boat problem

A boat is capable of moving with the speed v’ in still water. A river flows at the speed of u. If the boat moves in the river, with what speed and in which direction does it end up moving?

We use relative velocities to solve the problem. Consider a frame of reference moving with the river. Its velocity is u as seen from the ground. In this frame, the water is not moving. The speed of the boat in this frame is therefore, v’. Let’s say that the boat is pointed in a direction θ, its velocity vector being v’.

Figure 5

v’ is the velocity of the boat in the moving frame, the frame moves with the velocity u, so the velocity of the boat as seen from the ground is : v = v’ + u.

The boat ends up moving in the direction of v. The boat is kept pointed at the angle θ but it moves along v. Shown below are the boat’s positions at different instants of time.

Figure 6

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