Question

A muon formed high in Earth's atmosphere travels towards Earth at a speed

v = 0.970c

for a distance of 4.30 km as measured by an observer at rest with respect to Earth. It then decays into an electron, a neutrino, and an antineutrino.

(a) How long does the muon survive according to an observer at rest on Earth?


(b) Compute the gamma factor associated with the muon.


(c) How much time passes according to an observer traveling with the muon?


(d) What distance does the muon travel according to an observer traveling with the muon?


(e) A third observer traveling toward the muon at c/2 measures the lifetime of the particle. According to the observer, is the muon's lifetime shorter or longer than the lifetime measured by the observer at rest with respect to Earth? Explain.

Answer 1

given

V = 0.97 C

L_p = 4.3 km = 4300 m

a )

the time t = L_p / V

t = 4300 / 0.97 X 3 X 10⁸

t = 1.477 X 10^-5 Sec

b )

gamma factor = 1 / ( 1 - ( V / C )² )^1/2

= 1 / ( 1 - ( 0.97 C / C )² )^1/2

= 1 / ( 1 - ( 0.97 )² )^1/2

= 4.1134

c )

t_p = t /

t_p = 1.477 X 10^-5 / 4.1134

t_p = 3.59 X 10^-6 s

d )

distance d = V X t_p

d = 0.97 X 3 X 10⁸ X 3.59 X 10^-6

d = 1044.894 m

d = 1.044894 km

e )

third observer is moving in opposite direction to muon

so according to third observer speed of muon is higher than the original speed

then can say the muon's life is higher than that measured by observer when he is at rest