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John,
You seem to be solving a different problem.
There's is an inertial
reference frame O in which the two spaceships, one in front of the
other, are initially at rest.
There is a rope stretched from the tail
of the spaceship in front (point A) to the nose of the other spaceship
(point B).
At time zero in that frame both spaceships start
accelerating.
The way I read the problem, the spacecraft are stipulated
to accelerate in such a manner that the projected (onto frame O)
separation of point A and point B never changes.
You have them
accelerating such that an accelerometer on the tail of the lead
spaceship always has the same reading as an accelerometer on the nose of
the trailing spacecraft.
That's a different problem.
By the way, the rope breaks.
Consider a taut horizontal rope segment of
length L with the sun directly overhead. It is casting a shadow of
length L. A person rotates the rope about a horizontal axis that is
perpendicular to the rope while at the same time stretching the rope so
that its shadow remains at length L. The rope breaks.
How about having a bunch of identical rocket motors distributed along the rope?