OK, rapidity is one man's trick. Let me interject my favorite
parallel velocity transformation trick, which I like much better.
Consider three particles, 1, 2, and 3, which have velocities
which are parallel to one another. We will call these velocities
v12 (the velocity of 1 with respect to 2's rest frame), etc.
Let beta12 = v12/c
Galilean relativity tells us that beta12 + beta23 + beta31 = 0 .
(This also works for the nonparallel case. It is the familiar
relative velocity triangle formula.) This doesn't work for high
velocities. What does?
It turns out that there is a simple symmetrical formula which
contains the relativistic correction to the Galilean formula.
It was discovered by Eddington and never became popular, but it
should appeal to some others out there.
The reason I don't like the usual formula (and rapidity) is
because both treat some particle or frame in a special way; they
are not symmetrical, while the Galilean form is as I've written
it above. The relativistic form evidently reduces to the
Galilean form in the limit betajk << 1. (It can also be written