A muon is a lepton that is a higher-mass (rest mass 105 MeV/c^2) sibling to the

Question

A muon is a lepton that is a higher-mass (rest mass 105 MeV/c^2) sibling to the electron. Muons are produced in the upper atmosphere when incoming cosmic rays collide with the nuclei of gas molecules. The muon half-life is 1.5 microsecond, but atmospheric muons typically live much longer than this because of time dilation, as we saw in Chapter 27 in the textbook. Suppose 100000 muons are created 120 microseconds above the surface of the earth, each with kinetic energy 14 GeV . Assume that the muons don’t lose energy but move at a constant velocity directed straight down toward the surface of the earth. How many muons survive to reach the surface? A muon is a lepton that is a higher-mass (rest mass 105 MeV/c^2) sibling to the electron. Muons are produced in the upper atmosphere when incoming cosmic rays collide with the nuclei of gas molecules. The muon half-life is 1.5 microsecond, but atmospheric muons typically live much longer than this because of time dilation, as we saw in Chapter 27 in the textbook. Suppose 100000 muons are created 120 microseconds above the surface of the earth, each with kinetic energy 14 GeV . Assume that the muons don’t lose energy but move at a constant velocity directed straight down toward the surface of the earth. How many muons survive to reach the surface?

Explanation / Answer

K = (gamma - 1) m c^2

==> K = (1/(1 - v^2)^0.5 - 1) m c^2

==>14e9 = (1/(1 - v^2)^0.5 - 1) * 105e6

==> v =0.99997229179

c

==> gamma = 1/(1 - v^2)^0.5 = 1/(1 - 0.999972291792 ^2)^0.5 = 134.333333333

==> L = Lp/gamma = 120/134.33 = 0.893300249 km

==> t = L/v = 0.893300249 e3/(0.99997229179*3e8) = 0.00000297775 s

==> the survavial fraction of muons = e^(-t/T) = e^(-0.00000297775 /1.5e-6) = 0.13735771881

==> N = 0.13735771881 * 100000 =

muon

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