Chapter 26, Problem 9P

A muon formed high in Earth's atmosphere travels toward Earth at a speed v = 0.990c for a distance of 4.60 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 this observer, is the muon’s lifetime shorter or longer than the lifetime measured by the observer at rest with respect to Earth? Explain.

To determine

The lifetime of the muon according to an observer at rest on Earth.

Answer to Problem 9P

The lifetime of the muon according to an observer at rest on Earth is $1.55\times {10}^{-5}\text{s}$.

Explanation of Solution

The speed of muon is $0.990c$.

The distance covered by muon  is $4.60\times {10}^3\text{m}$.

Speed of light is $3.00\times {10}^8\text{m/s}$.

Formula to calculate the lifetime of the muon according to an observer at rest on Earth is,

$\Delta t=\frac{d}{v}$

• $\Delta t$ is the lifetime of the muon with respect to the observer at rest on Earth
• $v$ is the speed of muon
• $d$ is the distance covered by the muon
• $c$ is the speed of the light

Substitute $4.60\times {10}^3\text{m}$ for $\Delta t$ , $3.00\times {10}^8\text{m/s}$ for $c$ and $0.990c$ for $v$ to find $\Delta t$.

$\begin{array}{c}\Delta t=\frac{4.60\times {10}^3\text{m}}{0.990c}\\ =\frac{4.60\times {10}^3\text{m}}{0.990\left(3.00\times {10}^8\text{m/s}\right)}\\ =1.55\times {10}^{-5}\text{s}\end{array}$

Thus, the lifetime of the muon according to an observer at rest on Earth is $1.55\times {10}^{-5}\text{s}$.

Conclusion:

The lifetime of the muon according to an observer at rest on Earth is $1.55\times {10}^{-5}\text{s}$.

To determine

The gamma factor associated with the muon.

Answer to Problem 9P

The gamma factor associated with the muon is $\text{7}\text{.09}$.

Explanation of Solution

Formula to calculate the gamma factor associated with the muon is,

$\gamma =\frac{1}{\sqrt{1-{\left(\frac{v}{c}\right)}^2}}$

• $\gamma$ is the gamma factor associated with the muon

Substitute $0.990c$ for $v$ to find $\gamma$.

$\begin{array}{c}\gamma =\frac{1}{\sqrt{1-{\left(\frac{0.990c}{c}\right)}^2}}\\ =7.09\end{array}$

Thus, the gamma factor associated with the muon is $\text{7}\text{.09}$.

Conclusion:

The gamma factor associated with the muon is $\text{7}\text{.09}$.

To determine

The time interval measured by an observer travelling with the muon.

Answer to Problem 9P

The time interval measured by an observer travelling with the muon is $\text{2}\text{.19}\times {\text{10}}^{-6}\text{s}$.

Explanation of Solution

Given Info:

Formula to calculate the time interval measured by an observer travelling with the muon is,

$\Delta t_p=\frac{\Delta t}{\gamma }$

• $\Delta t_p$ is the time interval measured by an observer travelling with the muon

Substitute $7.09$ for $\gamma$ and $1.55\times {10}^{-5}\text{s}$ for $\Delta t$ to find $\Delta t_p$.

$\begin{array}{c}\Delta t_p=\frac{1.55\times {10}^{-5}\text{s}}{7.09}\\ =2.19\times {10}^{-6}\text{s}\end{array}$

Thus, the time interval measured by an observer travelling with the muon is $\text{2}\text{.19}\times {\text{10}}^{-6}\text{s}$.

Conclusion:

The time interval measured by an observer travelling with the muon is $\text{2}\text{.19}\times {\text{10}}^{-6}\text{s}$.

To determine

The distance travelled by the muon with respect to an observer travelling with muon.

Answer to Problem 9P

The distance travelled by the muon with respect to an observer travelling with muon is $650\text{m}$.

Explanation of Solution

The muon is at rest with respect to the observer moving with the muon. However, the observer sees the Earth moving a distance towards the muon.

Formula to calculate the distance travelled by the muon with respect to an observer travelling with muon is,

$d\text{'}=v\Delta t_p$

• $d\text{'}$ is the distance travelled by the muon with respect to an observer travelling with muon

Substitute $\text{2}\text{.19}\times {\text{10}}^{-6}\text{s}$ for $\Delta t_p$ , $3.00\times {10}^8\text{m/s}$ for $c$ and $0.990c$ for $v$ to find $d\text{'}$.

$\begin{array}{c}d\text{'}=\left(0.990c\right)\left(\text{2}\text{.19}\times {\text{10}}^{-6}\text{s}\right)\\ =\left(0.990\left(3.00\times {10}^8\text{m/s}\right)\right)\left(\text{2}\text{.19}\times {\text{10}}^{-6}\text{s}\right)\\ =650\text{m}\end{array}$

Thus, the distance travelled by the muon with respect to an observer travelling with muon is $650\text{m}$.

Conclusion:

The distance travelled by the muon with respect to an observer travelling with muon is $650\text{m}$.

To determine

The life time of the muon measured by the third observer with respect to the observer at rest on Earth.

Answer to Problem 9P

The third observer measures the muon's lifetime as longer than the lifetime measured by the observer at rest with respect to Earth.

Explanation of Solution

As the third observer travels toward the incoming muon, his speed relative to the muon is greater than that the speed measured by the observer at rest on Earth. Thus, the third observer’s observed gamma factor is higher.

Thus, the third observer measures the muon's lifetime as longer than the lifetime measured by the observer at rest with respect to Earth.

Conclusion:

The third observer measures the muon's lifetime as longer than the lifetime measured by the observer at rest with respect to Earth