Relative Relativistic Velocities

[Chinkylee]

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 can't find anything so confused now.

 

[ManyNames]

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]

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

 

[Dale]

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

 

[Fredrik]

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]

I see. I used the labeling BCA instead of CBA. This is of course just as correct, but more complicated then necessary.