(A2-13) ① What we know is that frame C1 is moving relative to N with a velocity of +0.5c, whereas Cormorant - now in C2 - is moving relative to N with a velocity of -0.5c. Using the relativistic velocity addition formula we can now find Cormorant's new velocity in frame C1.
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So where classical physics would have told us that Cormorant would now be moving at the speed of light in frame C1, we find - just like we did in section A2-4 - that relativity restricts him to a sub-light speed.
② At the beginning of the second leg:
Cormorant is at Mars and his position in frame C1, xC1(C) = 0. From ①, we now know that his new velocity in C1 is -0.8c.
Nefertiti is still on Earth so her position in frame C1, xC1(N) = -4.33 (we know this from A2-11) and his velocity in C1 continues to be -0.5c.
We are, for the moment, working within a single frame of reference (C1), so there are no relativistic complications to worry about. We can just use classical mechanics:
For Cormorant: starting point x = 0
distance covered = velocity x time = -0.8c x t
=> position after time t = -0.8ct
For Nefertiti: starting point x = -4.33
distance = -0.5c x t
=> position after time t = -4.33 - 0.5ct
At the time when they meet up again (tm), their distances from the origin of C1 will be the same, so:
-0.8ctm = -4.33 -0.5ctm
Which we can very easily solve, to find: tm = 14.43 minutes.
So 14.43 minutes is the time spent, in reference frame C1, by both Nefertiti and Cormorant during the second leg of the journey.
③ Anyone who happened to be stationary in frame C1 would be a moving observer of Nefertiti (stationary in N) and of Cormorant (stationary in C2), so we need to use the time dilation formula to find the times that each of them would actually experience:
Notice that the value obtained for Cormorant confirms what you already got for his second leg time in question A2-11.
The completed table you now get is:
④ We found, in ②, that14.43 minutes is the time spent, in reference frame C1, by both Nefertiti and Cormorant during the second leg of the journey.
At the start of this leg, the positions of Cormorant and Nefertiti were xC1(C) = 0 and xC1(N) = -4.33. So after 14.43 minutes travelling, their new positions will be:
xC1(C) = 0 + velocity x time = -0.8 x 14.43 = -11.54 light minutes
xC1(N) = -4.33 + velocity x time = -4.33 – (0.5 x 14.43) = -11.54 light minutes
So, as they must be, Cormorant and Nefertiti are now at the same point in space. Since, however, this is measured in the C1 reference frame, which is moving relative to Nefertiti at a speed of 0.5c, the distance travelled in her reference frame will be contracted. Using the usual length contraction equation, we get:
This is exactly as it should be: according to Nefertiti, she has not moved but during the two legs of the journey frame C1 has moved by +5 light minutes during each leg of the mission, so she should be at -10 light minutes relative to the origin of C1, at the end. Exactly as we have found. So, in spite of the complexity of dealing with all the relativistic corrections on each stage of this mission, it all fits together.
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