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