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Some tangential observations:
A less-important point is that they will disagree as to
what time it is. If at t=0 their clocks were synchronized,
then during the later rest period they will be offset by a
constant. This offset does not affect the dynamics during
the rest phase; we can just pretend they are living in
different "time zones". This is, however, a reminder of
some tricky dynamics that happened earlier, during the acceleration.
3) In addition to knowing the right answer, I would like to
understand what went wrong in my incorrect analysis.
I think it can be described roughly as a pole-in-barn
mistake. I correctly calculated the length of the barn, but
wrongly passed that off as the length of the pole (i.e.
rope) that was passing through the barn.
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