(2-11) ① u is the relative speed of the reference frames of the muons and of Nefertiti on the ground: for Nefertiti it is the speed at which the muons are approaching the ground whereas if you are a muon, it is the speed at which the ground is moving towards you. The key thing is that its value is the same whichever of the two frames it is measured in. Now, we have seen that in the muons’ frame of reference, the time elapsed during their journey to the ground is shorter than when it is measured from the ground. What is the only way you reconcile these things?
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② You therefore now need to distinguish the distances measured in the two frames of reference. To avoid (temporarily) getting mixed up with our 0s and 1s, let’s call the muons’ transit time tM as they experience it themselves and tN as Nefertiti measures it. Correspondingly, let’s call the distance covered, from Cormorant’s height to the ground, DM for the muons and DN, as Nefertiti would measure it.
Equating the velocities as measured in the muons’ frame and in Nefertiti’s:
Use this, together with the time dilation equation, to derive an expression for DM in terms of DN and u.
③ Now you are going to generalise this result, to find a relationship between the length of an object in the reference frame where it is static (D0) and in a frame where it is moving (D1). You need to be a bit careful with the nomenclature here. Remember that we chose to define the time measured in a muon’s reference frame as t0 because this is the rest frame for the time measurement derived from decay data. However, if we imagine a line between Cormorant and the ground, this line is moving in the muons’ reference frame, so the length of the line, as measured in this reference frame, is the length of a moving object and we therefore label it D1. So, confusingly, tM = t0 but DM = D1.
Likewise, in Nefertiti’s frame of reference, the time was dilated because the decaying muons – which we used as our clock - were moving in this frame. On the other the line between Nefertiti and Cormorant is fixed, so Nefertiti’s is the rest frame for the distance measurement: hence tN = t1 but DN = D0. but it is a moving frame as far as the time measurement is concerned, so this time is t1.
Use these considerations, together with your results from ②, to write a general expression for the length of an object in a reference frame where it moving, compared to its own static reference frame.
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