One of the many fundamental particles in nature is the muon mu. This particle ac

Question

One of the many fundamental particles in nature is the muon mu. This particle acts very much like a "heavy electron." It has a mass of 106 MeV/c^2, compared to the electron's mass of just 0.511 MeV/c^2. We are using E = mc^2 to obtain the mass in units of energy and the speed of light c). Unlike the electron, though, the muon has a finite lifetime, after which it decays into an electron and two very light particles called neutrinos (v). We'll ignore the neutrinos throughout this problem. If the muon is at rest, the characteristic time that it takes it to decay is about 2.2 mu s (tau_mu = 2.2 times 10^-6 s). Most of the time, though, particles such as muons are not at rest and, if they are moving relativistically, their lifetimes are increased by time dilation. In this problem we will explore some of these relativistic effects. Let's begin by considering some muons moving at various speeds relative to a stationary observer. If a muon is traveling at 70% of the speed of light, how long does it take to decay in the observers rest frame (i.e., what is the observed lifetime tau_mu of the muon)? Express your answer in microseconds to two significant figures. If a muon is traveling at 99.9% the speed of light, how long will it take to decay in the observers rest frame (i.e., what is the observed lifetime tau_mu of the muon)? Express your answer in microseconds to two significant figures.

Explanation / Answer

Given

decay time of the muon is T0 = 2.2*10^-6 S

Part A

muon moving with speed of 70% of speed of light that is 0.7*c

using the formula for time dilation T = T0 /sqrt(1-v^2/c^2)

                   = 2.2*10^-6/(sqrt(1-0.7^2*c^2/c^2)

                   = 2.2*10^-6/(sqrt(1-0.7^2) s

                   = 3.08062*10^-6 s

Part B

if the muon movingwith 99.9% of speed of light

      
       T = T0 /sqrt(1-v^2/c^2)

                   = 2.2*10^-6/(sqrt(1-0.999^2*c^2/c^2)

                   = 2.2*10^-6/(sqrt(1-0.999^2) s

                   = 4.92058*10^-5 s

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