Relative velocities in special relativity

[Kennedy]

Homework Statement

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

Homework 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

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.

 

[phinds]

Two spaceships fly toward a space station as shown in the figure.

What figure is that?

 

[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.

 

[Orodruin]

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.

Please show us what you tried.

 

[Kennedy]

Please show us what you tried.

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.

 

[PAllen]

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.

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.

 

[PAllen]

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

 

[Orodruin]

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.

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.

 

[Kennedy]

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

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?

 

[phinds]

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?

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.

 

[Kennedy]

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.

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?

 

[phinds]

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?

"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"

 

[Kennedy]

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.

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

 

[phinds]

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

Right.

 

[Kennedy]

"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"

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

 

[phinds]

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

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")

 

[Kennedy]

Right.

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))

 

[Kennedy]

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")

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.

 

[phinds]

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.

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?

 

[PeroK]

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.

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 $0.8c$?

 

[phinds]

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 $0.8c$?

Excellent idea. I think this might resolve some of his confusion.

 

[PAllen]

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))

Correct.

 

[Kennedy]

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 $0.8c$?

Well, then A approaches B at 8 m/s, and A would also approach the spaceship at 8 m/s. In the reference frame of A this means that spaceship B is moving at 8 m/s relative to A, and the space station also moves at 8 m/s towards A, relative to A.

Similarily, in the reference frame of B... A approaches at 8 m/s relative to B, and I would be looking for the speed at which the space station approaches B, in such a way that it is equal to 8 m/s... but doesn’t that mean that the speed of B would be the same as A, 8 m/s?

 

[PeroK]

Well, then A approaches B at 8 m/s, and A would also approach the spaceship at 8 m/s. In the reference frame of A this means that spaceship B is moving at 8 m/s relative to A, and the space station also moves at 8 m/s towards A, relative to A.

Similarily, in the reference frame of B... A approaches at 8 m/s relative to B, and I would be looking for the speed at which the space station approaches B, in such a way that it is equal to 8 m/s... but doesn’t that mean that the speed of B would be the same as A, 8 m/s?

An interesting answer! It might be worth sorting out your knowledge of reference frames and relative velocities in a non-relativistic setting before seriously tackling SR.

I would say that the speed of B must be $4m/s$. I think you need to analyse why that is the case.

 

[Kennedy]

An interesting answer! It might be worth sorting out your knowledge of reference frames and relative velocities in a non-relativistic setting before seriously tackling SR.

I would say that the speed of B must be $4m/s$. I think you need to analyse why that is the case.

I understand why the answer would be for B to travel at 4 m/s, but I don’t understand how I would have arrived at the answer myself.

If I take the reference frame of B, where A and the space station are moving in that reference frame, A is moving towards B at 8 m/s, and the space station is moving towards B at the same speed as B, which needs to be equal to the speed at which A is traveling towards B (in the reference frame of B, that would be 8 m/s)...

But then, if I take the reference frame of A, B is moving away from A at the same speed as it’s traveling relative to A (velocity of A - velocity of B), and the space station is moving at the same speed as A, towards A.

Do I have to take into account more than one reference frame to solve the problem?

 

[Ray Vickson]

I understand why the answer would be for B to travel at 4 m/s, but I don’t understand how I would have arrived at the answer myself.

If I take the reference frame of B, where A and the space station are moving in that reference frame, A is moving towards B at 8 m/s, and the space station is moving towards B at the same speed as B, which needs to be equal to the speed at which A is traveling towards B (in the reference frame of B, that would be 8 m/s)...

But then, if I take the reference frame of A, B is moving away from A at the same speed as it’s traveling relative to A (velocity of A - velocity of B), and the space station is moving at the same speed as A, towards A.

Do I have to take into account more than one reference frame to solve the problem?

Well... what do YOU think? You are given some initial data in the rest-frame of the space station, and are then asked about what would be seen from (the rest-frame of) B.

 

[Kennedy]

Well... what do YOU think? You are given some initial data in the rest-frame of the space station, and are then asked about what would be seen from (the rest-frame of) B.

So, yes... That becomes obvious because they’re asking about more than one reference frame in the problem. Okay, so to solve this without considering the aspect of special relativity, I need to look at both the reference frame of A and B, because if B were traveling at the same speed as A, than in the reference frame of A, the relative velocity of B would be zero, and likewise for in the reference frame of B.

 

[PeroK]

I understand why the answer would be for B to travel at 4 m/s, but I don’t understand how I would have arrived at the answer myself.

If I take the reference frame of B, where A and the space station are moving in that reference frame, A is moving towards B at 8 m/s, and the space station is moving towards B at the same speed as B, which needs to be equal to the speed at which A is traveling towards B (in the reference frame of B, that would be 8 m/s)...

But then, if I take the reference frame of A, B is moving away from A at the same speed as it’s traveling relative to A (velocity of B - velocity of A), and the space station is moving at the same speed as A, towards A.

Do I have to take into account more than one reference frame to solve the problem?

You need to find a way to visualise what is happening - and back up that visualisation with the appropriate calculations.

Since your homework was SR, here's my solution to the non-relativistic case:

Let the speed of B be $v$ in the space-station frame. The speed of A is $8m/s$ in this frame. Now, in B's reference frame:

The speed of A is $8m/s -v$ and the speed of the spacestation is $v$. We need these to be equal, hence:

$8m/s - v = v$ and $v = 4m/s$

(Note that, in any case, I would have a diagram showing A, B and the spacestation and the known and unknown velocities in the original frame. And then another diagram for B's frame.)

It is important to realize that your difficulties in this case were not with SR, per se, but with reference frames. I would really work on this, otherwise SR is going to be very tricky - or, at leasty, even trickier than it need be!