Relativity: Adding Velocities

When people first learn that nothing can exceed the speed of light, they often ask what would happen if two ships which were each going 75% of the speed of light happened to pass each other, head on. Wouldn't the people on each ship think the other ship was going 150% the speed of light?

Equivalently, some ask what if I see a ship moving 75% the speed of light with respect to me, and that ship fires a projectile forward at 75% the speed of light with respect to itself. Won't the projectile be moving at 150% the speed of light with respect to me?

A rocket launches another rocket (John D. Norton, U. Pittsburgh)

The answer is that velocities don't add in relativity. Instead, a concept called rapidity adds. The rapidity is the hyperbolic tangent of the velocity as a fraction of the speed of light. (Wikipedia has a good explanation.)

This calculator lets you put in velocities for the two ships and it computes the sum of their velocities. For example, in the case of two ships going 75% the speed of light, it shows that they'll observe each other moving 96% the speed of light.