From a train moving at speed , throw a ball forward at speed . By the Galilean transformation, the ball's speed seen from the ground would be .
That addition has no ceiling. Throw at from a train at and you get , a body moving faster than light.
The speed of light is in every frame, and no body can pass . A way of combining speeds that agrees with both comes out of the Lorentz transformation.
In the train frame the ball's speed is ; in the ground frame it is . Rewrite and with the reverse Lorentz transformation.
,
On dividing, cancels. Divide numerator and denominator by and use , and this is left:
This is the velocity addition rule. The numerator is the Galilean sum unchanged; only the denominator differs.
The denominator is larger than , and larger the larger and are. Throw at from a train at : the numerator is , the denominator , and .
The figure fixes the train at and steps the ball's speed from to in steps of . rises through , , , and , and stops at .
Working out gives , which is positive when and are both below . The combined speed never passes either.
Put . The numerator is and the denominator is . Dividing gives : the train's speed drops out.
Light sent from a moving train travels at on the ground too. The addition rule agrees with the constancy of the speed of light.
In the figure the train's speed changes through , and , but the answer in the lower row stays at .
If and are both much smaller than , is almost and .
Throw a ball at km/h from a bullet train at km/h and the relative difference from plain addition is only .
The figure combines two equal speeds. For each, addition and the rule both give . For each, addition gives and the rule ; for each, and .
The Newtonian addition of speeds is the rule with set to .