# Special Relativity: Relative velocity

• PeterPeter

## Homework Statement

I know that if you have 2 observers, "A", one at rest and the other "B" moving wrt "A" and if the moving observer shoots a projectile, then we can calculate the velocity of the projectile wrt "A" using the standard equation shown below.

I was thinking what equation would one use in this case: Two spaceships, A and B approach Earth at different velocities wrt the earth. What is the velocity of A wrt B?

## Homework Equations

u = (v+u')/(1+v u'/c^2)

## The Attempt at a Solution

Would u= velocity of A wrt Earth and v= velocity of B wrt Earth and u' be the velocity of A wrt B?

u = (v+u')/(1+v u'/c^2)

As reckoned from Earth's frame of reference, it would simply be the difference of the two individual velocities reckoned relative to the earth.

Chet

Would u= velocity of A wrt Earth and v= velocity of B wrt Earth and u' be the velocity of A wrt B?

u = (v+u')/(1+v u'/c^2)
No. How did you get that?

You would use the standard formula, but v and u' would be the speeds of A and B with respect to the earth. u gives the relative velocity.

You might find it useful to express the standard formula in this form:
$$V_{a/c} = \frac{V_{a/b} + V_{b/c}}{1 + (V_{a/b} V_{b/c})/c^2}$$

Hint: Let one ship be A and the other be C; let B be the earth. (Careful with signs.) Va/c means the velocity of a with respect to c, etc.

Your equation is right but you need to redefine the various velocities.

You are trying to measure u, the velocity of an object moving with velocity u' as measured in the primed frame, as seen from the unprimed frame. u is velocity of B as seen by A, v is the velocity between the primed and unprimed frames. Your equation reflects this but your definitions of u and u' contradict it.

The unprimed (observer) frame is in the A spaceship and the primed frame is on Earth. Then u' is the velocity of B spaceship wrt Earth and v is the relative velocity between the two frames. So u, u' and v are all negative.

So for example if A approaches Earth from the left and B from the right, v is negative, u' is negative and so u is negative also.