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http://math.ucr.edu/home/baez/physics/Relativity/SR/velocity.html
How Do You Add Velocities in Special Relativity?
Suppose an object A is moving with a velocity v relative to an object B, and B is moving with a velocity u (in the same direction) relative to an object C. What is the velocity of A relative to C?
v
u -------> A
-------> B
C w
----------------->
In non-relativistic mechanics the velocities are simply added and the answer is that A is moving with a velocity w = u+v relative to C. But in special relativity the velocities must be combined using the formula
u + v
w = ---------
1 + uv/c2
If u and v are both small compared to the speed of light c, then the answer is approximately the same as the non-relativistic theory. In the limit where u is equal to c (because C is a massless particle moving to the left at the speed of light), the sum gives c. This confirms that anything going at the speed of light does so in all reference frames.
This change in the velocity addition formula is not due to making measurements without taking into account time it takes light to travel or the Doppler effect. It is what is observed after such effects have been accounted for and is an effect of special relativity which cannot be accounted for with newtonian mechanics.
The formula can also be applied to velocities in opposite directions by simply changing signs of velocity values or by rearranging the formula and solving for v. In other words, If B is moving with velocity u relative to C and A is moving with velocity w relative to C then the velocity of A relative to B is given by
w - u
v = ---------
1 - wu/c2
Notice that the only case with velocities less than or equal to c which is singular is w = u = c which gives the indeterminate zero divided by zero. In other words it is meaningless to ask the relative velocity of two photons going in the same direction.
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