Lifetime of the Speeding Muon One of the many fundamental particles in nature is

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Lifetime of the Speeding Muon

One of the many fundamental particles in nature is the muon . This particle acts very much like a "heavy electron." It has a mass of 106MeV/c2, compared to the electron's mass of just 0.511MeV/c2. (We are using E=mc2 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 (). 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 2.2s(=2.2×106s). 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.

Let's begin by considering some muons moving at various speeds relative to a stationary observer.

Part 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 of the muon)?

Express your answer in microseconds to two significant figures.

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Part 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 of the muon)?

Express your answer in microseconds to two significant figures.

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A stream of particles, often called cosmic rays, is constantly raining down on the earth from outer space. (Figure 1) Most cosmic-ray particles are protons. When they crash into the upper atmosphere, they can convert into particles called pions (), which subsequently decay into muons. These muons can then continue toward the earth until they, too, decay. Let us consider the effects of time dilation on the cosmic rays.

Suppose that a cosmic-ray proton crashes into a nitrogen molecule in the upper atmosphere, 45 km above the earth's surface, producing a pion that decays into a muon. Assume that the muon has a downward velocity of 99.9943% the speed of light.

Part C

How far (d) would the muon travel before it decayed, if there were no time dilation?

Express your answer in meters to three significant figures.

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Part D

Now, let us consider the effects of time dilation. How far would the muon travel, taking time dilation into account?

Express your answer in kilometers to two significant figures.

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Figure 1 of 1

Lifetime of the Speeding Muon

One of the many fundamental particles in nature is the muon . This particle acts very much like a "heavy electron." It has a mass of 106MeV/c2, compared to the electron's mass of just 0.511MeV/c2. (We are using E=mc2 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 (). 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 2.2s(=2.2×106s). 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.

Part 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 of the muon)?

Express your answer in microseconds to two significant figures.

=   s  

SubmitHintsMy AnswersGive UpReview Part

Part 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 of the muon)?

Express your answer in microseconds to two significant figures.

=   s  

SubmitMy AnswersGive Up

A stream of particles, often called cosmic rays, is constantly raining down on the earth from outer space. (Figure 1) Most cosmic-ray particles are protons. When they crash into the upper atmosphere, they can convert into particles called pions (), which subsequently decay into muons. These muons can then continue toward the earth until they, too, decay. Let us consider the effects of time dilation on the cosmic rays.

Suppose that a cosmic-ray proton crashes into a nitrogen molecule in the upper atmosphere, 45 km above the earth's surface, producing a pion that decays into a muon. Assume that the muon has a downward velocity of 99.9943% the speed of light.

Part C

How far (d) would the muon travel before it decayed, if there were no time dilation?

Express your answer in meters to three significant figures.

d =   m  

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Part D

Now, let us consider the effects of time dilation. How far would the muon travel, taking time dilation into account?

Express your answer in kilometers to two significant figures.

d =   km  

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Figure 1 of 1

Explanation / Answer

given    =   2.2*10^-6 s

v    = 0.7 * c

v /c =   0.7

from the time dilation equation

'     =   / sqrt ( 1 - (v/c)2)   

= / sqrt ( 1 - (0.7)2)

=   3.081*10-6 s

(b)

velocity of moun is

v = 0.999 * c

v/ c =   0.999

'     =   / sqrt ( 1 - (v/c)2)   

= / sqrt ( 1 - (0.99)2)

= 49.20*10-6 s

(c)

velocity of the moun is

v   =   0.999943*c

=   0.999943*3*10^8 m/s  

= 2.999829*10^8 m/s

d   =  vt

=    2.999829*10^8 m/s )( 2.2*10^-6 s )

= 6.5996238*10^2 m

(d)

v = 0.999943*c

v / c =   0.999943

from the time dilation

'     =   / sqrt ( 1 - (v/c)2)    

=  3.081*10-6 s / sqrt ( 1 - (0.999943)2)

= 206. 05*10^-6 s

d   =  vt

=  (  2.999829*10^8 m/s )( 206.05*10^-6 s )

= 61.81 km

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