In classical physics, velocities simply add. If an object moves with speed u in one reference frame, which is itself moving at v with respect to a second frame, the object moves at speed u+v in that second frame.
This is inconsistant with relativity because it predicts that if the speed of light is c in the first frame it will be v+c in the second.
We need to find an alternative formula for combining velocities. We can do this with the Lorentz transform.
Because the factor v/c will keep recurring we shall call that ratio β.
We are considering three frames; frame O, frame O' which moves at speed u with respect to frame O, and frame O" which moves at speed v with respect to frame O'.
We want to know the speed of O" with respect to frame O,U which would classically be u+v.
The transforms from O to O' and O' to O" can be written as matrix equations,
where we are defining the β's and γ's as
We can combine these to get the relationship between the O and O" coordinates simply by multiplying the matrices, giving
This should be the same as the Lorentz transform between the two frames,
These two sets of equations do look similar. We can make them look more similar still by taking a factor of 1+ββ' out of the matrix in (1) giving#
This will be identical with equation 2 if
Since the two equations must give identical results, we know these conditions must be true.
Writing the β's in terms of the velocities equation 3a becomes
which tells us U in terms of u and v.
A little algebra shows that this implies equation 3b is also true
Multiplying by c we can finally write.
Notice that if u or v is much smaller than c the denominator is approximately 1, and the velocities approximately add but if either u or v is c then so is U, just as we expected.