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I'm struggling to ascertain which reference frame (pion or detectors) has the 'longer time taken' in the following example:

Consider a pion which is travelling at 0.95c. It passes two detectors 34m apart.

If you were to calculate the time taken in the frame of reference of the detectors for the pion to travel between the two detectors, would the detectors be an external observer that would see the time taken dilated- i.e. the pion's clock appears to be running slow compared to the detector's clock?

Or would you have to consider the length contraction and not the time dilation? In what reference frame would the length be contracted? Would it be that, as the pion travels the 34m, the 34m gap is travelling 0.95c relative to the pion so the gap appears smaller?

If you use $t=t_{0}/\sqrt{(1-v^2/c^2)} $, where $t_{0} $ is the 'proper time' (from the frame of reference of the pion?), the answer will be ~380 ns whereas if just $t=s/v$ is used, the answer is ~120 ns. I would have thought the longer time is correct, but apparently the answer is in fact 120 ns.

What is the best way to tackle this problem?

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Whichever approach you use, the probability of the pion decaying between detectors should be the same. Whether observed in the pion frame, or the lab frame.

The pion thinks "that gap is not very big, and I'm going fast; it won't take me long".

The lab thinks "that pion is going fast; its clock is slowing down; it doesn't think it took very long".

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