Relative velocity between two accelerating observers

[dEdt]

Suppose two observers A and B are accelerating at the same rate with respect to an inertial reference frame. In A's frame, does B move at a constant velocity? I'm pretty sure (but not certain) that the answer is yes, but even if it is I can't think of a justification.

 

[StationZero]

does B move at a constant velocity?

The problem with your question is that you didn't specify a directional vector. I'm assuming you meant to specify that A and B are traveling parallel to one another in the same direction, and, if that is the case, then yes, both A and B would appear to be at rest in each others reference frame relative to one another. To an external observer, they would appear to be accelerating at the same rate, velocity plays no role in this scenario.

If their directional vector is not identicle, then you simply sum the accelerations to get their relative accelerations. For example, if A and B are moving in opposite directions and accelerating at the same rate, say 10m/s squared, then they are accelerating away from each other at 20 m/s squared.

 

[Bill_K]

No, they would not. To an observer on the ground, at any time, A and B will be seen to have the same velocity. However the motion of A and B affects their simultaneity. What appears to be simultaneous to the ground observer will no longer appear simultaneous to A, or to B. Once A has acquired a velocity v, he observes B "at the same time" to be farther along on its path and therefore traveling faster.

 

[StationZero]

A and B will be seen to have the same velocity

How do you get anything other than an instantaneous velocity out of an acceleration vector?

Once A has acquired a velocity v, he observes B "at the same time" to be farther along on its path and therefore traveling faster.

How do you get one body traveling faster than the other when they are accelerating at the same rate?

 

[Bill_K]

The original question was ambiguous, and I should have asked for a clarification. I agree if A and B are side by side there will be no difference. The usual picture is that A is in front of B, or vice versa, and that's what I was assuming. Similar conclusions hold about the distance between them. In the frame of the ground observer the distance remains the same. But in the frame of A or B, it does not.

 

[Nugatory]

How do you get one body traveling faster than the other when they are accelerating at the same rate?

(Making the same assumption about the direction of motion as Bill_K - if that's not what OP intended, then this post should be considered an interesting digression)

Bill_K's answer about the relativity of simultaneity is spot on. That phrase "accelerating at the same rate" is equivalent to "both changing their speed by the same amount at the same time" and those words in bold should be setting off alarm bells.

The only thing that I can add is that this problem is basically Bell's Spaceship Paradox... There's a very long thread recently, kinda went off the rails after a while, but has some links to good references near the start.

 

[StationZero]

Thanks Nugatory, I had another post in here that was relevant to the thread but it got deleted, just as several of my posts have been deleted. So, I don't think you got the entire picture here. In any case, I'm reading the forum guidlines now so I don't get any more infractions, please stay tuned.

 

[Nugatory]

I agree if A and B are side by side there will be no difference.

Are you sure? There are still RoS questions about when each ship observes the velocity changes of the other ship as compared with the ground observer. I'm asking not arguing here... playing with the problem myself now.