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Two separate suitably short but intense bunches of muons, "A" and "B", are both supposed to be constantly accelerating (in an otherwise sufficiently flat region) with constant proper acceleration vectors $\bf a$ and $\bf b$ (as measured by any system of participants who are mutually at rest) in the same direction and co-linearly (let's say "A leading in front", and "B trailing behind"), but not necessarily equally accelerated (therefore acceleration vectors $\bf a$ and $\bf b$ don't necessarily have equal magnitudes).

Meanwhile the muons of both bunches decay, with equal and constant proper mean lifetime $\tau_{\text muon}$, and correspondingly with equal proper "half-life duration Log[ 2 ] $\tau_{\text muon}$".

Also, the two bunches are separated such that they always "remain in sight" of each other; such that

  • B (or some suitable accompanying instrumentation) always finds that it takes 1 half-life from "stating a ping" until "observing A's reflection", while

  • A always finds that it takes 2 half-lives from "stating a ping" until "observing B's reflection".

Question:
What's the magnitude $| \bf b | $ ?
Can it be calculated at all, in terms of "mean lifetime $\tau_{\text muon}$" and "speed of light $c$", based on the described setup conditions?

(Surely the described setup conditions are consistent with bunches A and B being called "rigid to each other", or having been "ends of a rigid rod", in the sense of this question and that question and the various comments there.

My question is intended to obtain illustrations of different approaches to the "rigid rod" problem which were indicated in those comments; even if they obtain the same answer value.)

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