Muons are subatomic particles that can be generated in the upper atmosphere of t
ID: 2077585 • Letter: M
Question
Muons are subatomic particles that can be generated in the upper atmosphere of the earth due to cosmic ray interactions with the atoms of the atmosphere. The particles are measured to decay in 2.197 mu s as measured in their own reference frame. Suppose some muons are produces 10 km above the surface of the earth, as measured by an observer on the surface of the earth. a) Calculate the largest possible distance one of these particles could travel in a time of 2.197 mu s as measured from an observer on the surface of the earth. You will get a distance that is shorter than 10km, however many of these muons are actually measured to reach the surface of the earth. This means, as measured by an observer on the surface of the earth, these muons lived longer than 2.197 mus. Briefly explain how this can be. b) Suppose you observe a muon that is traveling at a speed of 0.999.c relative to the earth's surface. Calculate the distance this muon travels before decaying towards the earth according to an observer on the surface of the earth. You will get a distance that is larger than 10km. c) Calculate the distance traveled by the muon in part (b), in the muon's reference frame, for the 10 km long trip through the Earth's atmosphere.Explanation / Answer
A.
Muons are moving at relativistic speeds. So, according to Special theory of relativity, Time dialates for a moving particle, as observed from another inertial frame.
Also, distance(lengths) of moving particles gets shortened.
As, time dialates,d ecay will happen in a longer period ,as observed on earth frame, some muons will reach earth.
B.
v=0.999c
Time dialates so, t =to/(1-v2/c2)
t0 =2.197×10-6seconds
t= 2.197×10-6/(1-(0.999c)2/c2.
2.197×10-6/(1-.998)
= 2.197×10-6/0.0447
=29.015×10-6 seconds.
Distance travelled in this time = v×t.
Distance = 0.999×3×108×29.015×10-6
=86.957×102m
Or 8.6957km.
In muons frame, distance shortens by do×(1-v2/c2)
do× 0.0447=10×0.0447 = 0.447km.
Hence time it took is =
Distance/speed
447m/0.999c
= 149.15×10-8 seconds or 1.49us.
Which is less than decay time of muon.
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