Lifetime of the Speeding Muon?

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 electrons mass of just 0.511 MeV/c^2. We r usin E = mc^2 to obtain the mass in units of energy & the speed of light c. If the muon is @ rest, the characteristic time that it takes it to decay is about 2.2 mu s (tau mu = 2.2 x 10^-6 s).

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A) If a muon is traveling at 70% of the speed of light, how long does it take to decay in the observer's rest frame (i.e., what is the observed lifetime tau_mu of the muon)? =__MuS

B) If a muon is traveling at 99.9% the speed of light, how long will it take to decay in the observer's rest frame (i.e., what is the observed lifetime tau_mu of the muon)? =__ MuS

C) How far (d_mu) would the muon travel before it decayed, if there were no time dilation? = __m

D)How far would the muon travel taking time dilation into acct?

Answer 1

A)

τ _µ' = τ _µ · γ = τ _µ / √(1 - (v/c)²)

= 2.2×10⁻⁶s / √(1 - (0.7)²)

= 3.08×10⁻⁶s

B)

τ _µ' = 2.2×10⁻⁶s / √(1 - (0.99)²)

= 1.56×10⁻⁵s

C)

s = v · τ _µ

=>

at v/c=0.7

s = 0.7 · 3×10⁸m/s · 2.2×10⁻⁶s = 462m

at v/c=0.99

s = 0.99 · 3×10⁸m/s · 2.2×10⁻⁶s = 653.4m

D)

s = v · τ _µ'

=>

at v/c=0.7

s = 0.7 · 3×10⁸m/s · 3.08×10⁻⁶s = 646.8m

at v/c=0.99

s = 0.99 · 3×10⁸m/s · 1.56×10⁻⁵s = 4633.2m