(2-10)
The results (muons per square metre per second) are:
Cormorant: 12835 Nefertiti: 6001
① Ignoring time dilation, for the present, use the decay rate equation to deduce the “classical” time taken for the muons to move between Cormorant’s altitude and Nefertiti’s and hence deduce the velocity of the muons detected in this experiment. Explain the problem with this result.
② Now let’s bring in a bit of special relativity:
The time taken to reach the ground, as deduced from the amount of decay that occurred, is time as experienced by the muons (since they are the ones who get to decay). So this is t0, the time measured in the muons' static reference frame.
The distance between Cormorant and the ground is distance measured by an observer (Nefertiti) who – from her own point of view – is static on the ground.
Clearly, Nefertiti and the muons are moving relative to one another: these two numbers are therefore not measured in the same frame of reference and so cannot be used directly to calculate the velocity. Instead we have to take account of time dilation, to deduce t1, the time taken in Nefertiti’s frame of reference. Because Nefertiti is, in effect, a moving observer of the muons, t1 will be longer than the time measured in the muons’ own frame. So:
write down the equation that relates the time measured in Nefertiti’s frame (t1) to the time you have calculated, in the muons’ frame (t0).
③ At first sight this may look unhelpful, since we seem to have two unknowns, t1 and u (the velocity of the muons). A moment’s pause, however, will allow you to remember that u is equal to distance / time. We need to measure distance and time in the same frame of reference, so since we know the distance in Nefertiti’s frame (Cormorant is at a fixed height of 5000m relative to her), the time should also be measured in her reference frame (t1). So now recast your equation from ② to replace u with the appropriate expression and rearrange to find an expression for t1.
④ Now you can put in the numbers to calculate t1 and hence the velocity of the muons. Does this look better?