__Return to current section__**: ****Velocity Composition****.**

**Exercise: **Derive and interpret the relativistic **velocity composition law:**** **** ***U*′ = *ṼU*.

Therefrom, derive the corresponding composition laws for time dilations and relative velocities.

From the previous section on passive Lorentz transformations, the translation of coordinates of an event *X* from the unprimed inertial system to the primed system can be written as *X*′ = *ṼX*, where *V* is the proper velocity of the primed system with respect to the unprimed system. But that event may correspond to a moving particle. That particle has a history (a series of events observable by other systems) and a proper time parameterizing its history. We would take the derivative of its history with respect to its proper time to get its proper velocity *U*, whose coordinates can be calculated in any inertial system. Since *V* is a constant for these inertial systems, we can take the derivative of the last equation with respect to the particle's proper time to get *U*′ = *ṼU*.

As mentioned in deriving the Lorentz transformation in the previous section, because we are considering motion in one dimension (our x-axis), **v**∧**u** = **v**∧**u'** = 0, so simply multiply the terms and collect scalar and vector parts separately:

Similarly,

As a quick check on signs, if *v*/*c** *is small, notice these formulas reduce to Galileo's velocity addition formula **u** = **u'** + **v**.

__Return to current section__**: ****Velocity Composition****.**