A muon has a lifetime of 2.20 x10-6 s when at rest, after which time it decays into other particles.
a) Ignore any effects of relativity discussed in this section. If the muon was moving at 0.99c, how far would it travel before decaying into other particles, according to Newtonian mechanics?
b) How long would the muon last, according to an observer in the earth’s frame of reference who viewed the muon moving at 0.99c?
c) How far would the muon actually travel, when viewed moving at 0.99c?
d) Compare the two distances traveled. Explain why this type of evidence is excellent support for the theory of relativity.
2. Homework Equations
d = vt
Δtm = Δts / 1-√v2/c2
The Attempt at a Solution
d = (0.99c)(3.00 x108m/s)
d = 2.97 x10-8m/s
(2.97 x10-8m/s)(2.2 x10-6s)
= 653.4 m
b) Δtm = Δts / 1-√v2/c2
Δtm = 2.2 x106 / √1- 0.9801
Δtm = 1.56 x10-5 s
The muon will last 1.56 x10-5 s in the earth’s frame of reference, moving at 0.99c
c) Using special relativity, the muon would last for dSR= vτ, where τ is now the relativistic lifetime of the muon.
dSR= (0.99c)(3.00 x108m/s)(1.56 x10-5 s)
dSR= 4633.2 m
These results are a good demonstration of relativity because they show that time dilation becomes significant as the velocity approaches the speed of light.
I'm just not sure if I did b) and c) right, in c) when it says "How long will it last", I assume that means how many seconds, but I could be wrong. It seemed logical in c) to use the same equation as a) but simply change the value of τ to the relativistic value, but as I haven't had to use this equation before I'm not sure if it is the right one to be using. I would appreciate any feedback