# Homework Help: Relative velocities in special relativity

1. Dec 28, 2017

### Kennedy

1. The problem statement, all variables and given/known data
Two spaceships fly toward a space station as shown in the figure. Relative to the station, spaceship A has speed 0.8c. Relative to the station, what speed is required of spaceship B such that its pilot sees A and the station approach B at the same speed?

(a) 0.40c (b) 0.50c (c) 0.60c (d) 0.70c (e) 0.80c

2. Relevant equations
u = u' + v/(1+(v)(u')/(c^2)), the relative velocity formula for speeds near the speed of light
L = L0/ϒ, where ϒ = 1/(1-(v/c)^2))^(1/2), maybe

3. The attempt at a solution
I know that I can use the relative velocity equation to figure the relative speed of A according to B by substituting the speed of B in for v in the relative velocities equation. I want to get the relative speed of A to be the same speed for which the space station is approaching B, but I can't seem to derive a formula exactly for that. I'm confused about what I have to work with here.

2. Dec 28, 2017

### phinds

What figure is that?

3. Dec 28, 2017

### Kennedy

It wasn't a helpful figure. It just showed two spaceships in a line, A behind B, and the space station a short distance away from B. I could have drawn something similar myself. I don't think it was too helpful to the question.

4. Dec 28, 2017

### Orodruin

Staff Emeritus
Please show us what you tried.

5. Dec 28, 2017

### Kennedy

I tried setting u' in the relative velocity formula to 0.8c, the speed of ship A, and then trying to somehow find an equation that has the speed of the space station relative to B. But for some reason I take the space station to be at rest. Because we want the relative speed of ship A and the space station to be equal, I would set the expressions equal to each other, and solve for v, but since I'm not given the speed of the space station relative to anything else, I got stuck, and wasn't able to do much.

6. Dec 28, 2017

### PAllen

Actually, that is crucial. Other configurations would have no such solution, e.g. the rockets approaching either side of the station. Big hint - the signs in your application of velocity addition should make use of this configuration.

7. Dec 28, 2017

### PAllen

If B is moving at v relative to the station, at what speed is the station moving relative to B?

8. Dec 29, 2017

### Orodruin

Staff Emeritus
I am sorry, but this is just telling us about what you did, not showing us what you did. You need to start writing down what equations you are using, what exact assumptions you are making, and what results you get. Do not just assume that everyone has a velocity addition formula where things are called exactly what they are called in your reference or that we can see what is in your notes. By not writing things out explicitly you are just making it impossible for us to help you.

9. Dec 29, 2017

### Kennedy

Well then, the station would be moving at the same speed as B relative to B... is that right? So, in that case, I would be looking for a relative velocity of A that is the same speed as the spaceship B?

10. Dec 29, 2017

### phinds

Personally, I find this an extremely confusing and confused answer to a VERY simple question, with the irrelevant introduction of A. To repeat the simple question, "If B is moving at v relative to the station, at what speed is the station moving relative to B?" HINT: the answer can be given in one letter.

11. Dec 29, 2017

### Kennedy

Okay, so I took the relative velocity of spaceship A (according to B) to be equal to 0.8c + v/((1+((0.8c)(v)/(c^2))), where v is the speed of spaceship B, and the 0.8c is my relative velocity of A according to the space station, but since I assume that the space station is at rest, then 0,8c would just be the speed of A. My relative velocity of A (according to B) needs to be equal to the speed at which the space station is approaching spaceship B, but since I’m assuming that the space station is at rest, would that not just be the speed of B?

12. Dec 29, 2017

### phinds

"At rest" is a confusing and/or meaningless concept. You HAVE to specify "at rest relative to <something> " since there is no absolute "at rest"

13. Dec 29, 2017

### Kennedy

v, right. The space station is moving towards B at the same speed (v) as B is moving towards the space station.

14. Dec 29, 2017

### phinds

Right.

15. Dec 29, 2017

### Kennedy

Would it be fair to say that the space station is at rest relative to both spaceships?

16. Dec 29, 2017

### phinds

Better to say something like "I am considering the movement of the spaceships in the frame of reference of the space station" Everything is at rest in its own frame of reference (and that is not any kind of absolute "at rest")

17. Dec 29, 2017

### Kennedy

Okay, so the goal here would be to find a relative velocity of A according to B, such that the relative velocity of A is equal to the speed of spaceship B. So, I would set v = (0.8c - v)/(1 - (((0.8c)(v))/(c^2))

18. Dec 29, 2017

### Kennedy

Oh, so the question gives me the speed of the spaceship A in the reference frame of the space station, and then I’m considering everything to be in the reference frame of B when solving the problem, because everything is moving relative to the spaceship B.

19. Dec 29, 2017

### phinds

Everything is moving relative to everything (except itself) so I don't find this statement meaningful or helpful. Reread the question. In what frame of reference do you need the speeds required for the answer? What frame or frames do you need to use to get those speeds?

20. Dec 29, 2017

### PeroK

I wonder whether an idea might be to solve this problem first for non-relativistic velocities?

The solution process should be the same, although there are different formulas for velocity addition.

So, what if the speed of A was $8m/s$ rather than $0.8c$?