relativistic velocity addition
In everyday life velocities just add. If you walk at 5 km/h toward the front of a train moving at 100 km/h, the ground sees you at 105 km/h. Relativity says this simple sum is only an approximation, very good at low speeds but wrong near the speed of light. The correct rule bends the answer downward so that no combination of speeds ever reaches or passes c.
The formula does this with a single extra term in the denominator. Add two speeds the relativistic way and you always get something a little less than the naive total, and the closer the speeds are to c, the more the result is held back. Fire a bullet at half light speed from a ship already moving at half light speed, and the ground does not see light speed; it sees about 0.8 c, comfortably below the limit.
The truly elegant part is what happens with light itself. Plug in c for one of the speeds and the formula returns exactly c, no matter what the other speed is. This is not a coincidence patched in by hand; it is built into the structure so that the second postulate, the constancy of light, holds automatically in every frame. Light's speed is the fixed point the whole rule is designed around.
A homely way to see the spirit of it: the law of adding velocities is really the law for combining two Lorentz transformations, one stacked on the other. Just as two small rotations combine into one rotation rather than a bigger sliding shift, two boosts combine into a single boost whose speed is reined in below c. Speeds in relativity behave less like distances you pile up and more like angles that saturate.
Two velocities combine through this rule; the result is always below c, and any light speed input returns exactly c.
The simple sum u₁ + u₂ is just the low-speed limit of this formula; it is accurate to a tiny fraction at human speeds, which is why Galilean addition served us for centuries.