# Relative Relativistic Velocities

## Main Question or Discussion Point

Relative Relativistic Velocities!!!

Suppose 2 space ships are moving at 0.51c, one moving left, the other one moving right.
A<------- -------->B
0.51c 0.51c

the speeds measured from a stationary observer. What is the speed of B in A's frame of reference?

Been thinking about the answer but cant find anything so confused now.

Related Special and General Relativity News on Phys.org

Oh, i think you will find your answer by saying everything remains relative. I know that some people will state that you need to add the speeds up from both frames, but if this was to hold, both of them would find each other moving away from each other faster than light.

Meir Achuz
Homework Helper
Gold Member

The speed of B in A's frame of reference is given by
v=(.51+.51)c/(1+.51X.51).

Dale
Mentor

Google the phrase "relativistic velocity addition". This is well-known and there is a lot of material available.

Fredrik
Staff Emeritus
Gold Member

The velocity addition formula with c=1 is w=(u+v)/(1+uv). In this case, u=-0.51, w=0.51, and v is what we're trying to find. So solve for v: w+wuv=u+v, w-u=(1-wu)v, v=(w-u)/(1-wu)=0.81.

Clem's solution "w unknown, u=v=0.51" looked wrong to me, but I'm getting the same result using "v unknown, w=-u=0.51".

JesseM

The velocity addition formula with c=1 is w=(u+v)/(1+uv). In this case, u=-0.51, w=0.51, and v is what we're trying to find. So solve for v: w+wuv=u+v, w-u=(1-wu)v, v=(w-u)/(1-wu)=0.81.

Clem's solution "w unknown, u=v=0.51" looked wrong to me, but I'm getting the same result using "v unknown, w=-u=0.51".
Look at the way u, v and w are defined in terms of the diagram here, with A moving at v to the right relative to B and B moving at u to the right relative to C, and A moving at w to the right relative to C. Now relabel the spaceship on the left in the OP's diagram as "C", relabel the spaceship on the right as "A", and have the observer in the middle be "B", like so:

C<------- B -------->A

Then shift to C's rest frame, where both B and A are moving to the right, with A having a higher speed than B:

C ------->B ------------->A

...and this will then match the diagram on the page I linked to above, with B moving at u=0.51c to the right in C's frame, and A moving at v=0.51c to the right in B's frame.

Fredrik
Staff Emeritus