Question

The proper average (or mean) lifetime of a muon (a subnuclear particle) is t = 2.2

Answer 1

In order to answer this question, you need to know the time dilation equation:

with t₀ being the time in reference to the object moving at velocity v, t is the time observed for a stationary viewer, and c is the speed of light.

The relativistic length contraction equation is also needed:

with L₀ being the rest length observed, v neing the velocity of the object, and c being the speed of light.

The simple equation to get the distance d is for a given velocity v and time t also used:

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Part a) In order to find the lifetime of the muons in the laboratory frmae of reference, use the time dialtion equation, with the time for the object t₀ being 2.2 us, and the ratio v/c being 0.996:

So the muons last an average of 24.6 us in the laboratory frame of reference.

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Part b) To find out how far the muons travel in the laboratory frame, just use the distance equation, with the velocity v being 0.996 c, and the time being 11.2x10^-6 s (11.2 us):

So in the laboratory setting, the muons travel about 7360 m before they decay.

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Part c) The relative velocity between the laboratory and the muons is symmetric, so an observer traveling with the muons will see the lab going by at a velocity of 0.996c.

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Part d) In order to determine how far the lab goes in the frame of the muon before the muon decays, use the length contraction equation, with L₀ being the length covered by a stationary obsever (the 3344 m found earlier), and the ratio v/c being 0.996:

So in the muon's reference frame, about 657 m pass by in it's lifetime.