Okay. If I'm sitting on the back of a pickup truck that's moving backwards at 5 m/s, and throw a ball off the back of the truck at 3 m/s, how fast does the ball move relative to the ground? We know this intuitively: since the ball and the truck are moving in the same direction, the ball moves at 8 m/s. Similarly, if the truck and ball were moving in opposite directions, we would subtract the velocities to get the answer. Now then, on to the tricky bit. If our truck were somehow moving at half of the speed of light--denoted as 0.5c, and we were to throw a ball in the truck's direction of motion at 0.5c, the ball is now moving at c relative to the ground, right? Bzzt! Wrong answer! It is impossible—as we've demonstrated in relativistic momentum (q.v.)—for anything except light to move as fast as light. Therefore, a new equation for addition of relativistic velocities:

V = v1+v2/sqrt (1+ (v1·v2/c²) )

, where c is the speed of light and v1 and v2 are the velocities of the truck and ball. By plugging 0.5c in for both variables, we can see that even though both the truck and the ball are moving at 0.5c, the velocity of the ball is just 0.8c relative to the ground. If the truck and ball were both somehow moving at the speed of light, the velocity of the ball would be-- c!

Normal velocity addition is acceptable with smaller velocities, of course; with everyday velocities, the two methods are indistinguishable.