How is the universal average speed obtained?

Settings:

A swimmer is trying to swim across a river, in a straight line from a point A on the left bank to a point B on the right bank, and then will return to A following the same path. The swimmer’s speed in still water is c, and the water in the river is flowing at a speed of v.

Question:

What is his average speed in this round trip?

swimming settings

Main idea of the solution:

To compensate for the influence of the water, the swimmer’s swimming direction has to be different from his intended direction AB.

If we can get his upstream speed v1 and downstream speed v2, then the average speed can be easily obtained by

vavg 1

that is,

vavg 2

So our first step is to get v1 and v2.

Using pythagoras theorem, we can get v1 and v2. Then by putting them into the above equation, the average speed equation can be obtained:

general average speed

The details can be downloaded here