The boat swam for some time with the river and the same amount against the current.

The boat swam for some time with the river and the same amount against the current. Prove that it takes the same amount of time to swim the same distance in still water.

Let a be the speed of the boat and b the speed of the river. Downstream, as well as against the current, the boat sails for the same time t. Moreover, downstream its speed becomes equal to (a + b), and upstream – (a – b). Let us find the total distance the boat will sail, moving in the same time t both along the river and against it:

(a + b) * t + (a – b) * t = (a + b + a – b) * t = 2at.

Let us find the distance the boat will sail in the same time t in still water:

at + at = 2at.

That is, beyond both the river and in stagnant water, the boat under the given conditions will sail the same distance equal to 2at. Q.E.D.