## Special Relativity Demystified (1)

In a smooth running river, a gentleman is swimming. As this is his first time of swimming in a river, to make it safe, he is swimming along one bank of the river. He has started from a downstream point A, and intends to reach an upstream point B, and then returns to point A.

Here is the question: What is his average speed during this round trip, if his swimming speed in still water is c, and the river is running downstream at a speed of v?

It is easy to get the average speed, as can be seen from the following:

t_{AB} =\frac{s}{c-v}

t_{BA} =\frac{s}{c+v}

t_{sum} =t_{AB}+t_{BA}=\frac{s}{c-v}+\frac{s}{c+v}=2s\left ( \frac{c}{c^2-v^2} \right )

c_{avg} =\frac{2s}{t_{sum}}=\frac{c^2-v^2}{c}=c\left ( 1-\frac{v^2}{c^2} \right )

We can see that the average speed is lower than the swimmer's speed in still water.

Now let's change the scene a little bit. In a peaceful lake with still water, a long boat is running at a constant speed in a straight line, and the same person is swimming along the side of the boat in round trips. What is his average speed this time?

Our formula still works, because relative motion only cares about the relative position between the two objects. As for which one is moving, it does not matter at all.