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Chapter 26: Problem 23

The mean (average) lifetime of a muon in its rest frame is $2.2 \mu \mathrm{s}\(. A beam of muons is moving through the lab with speed \)0.994 c .$ How far on average does a muon travel through the lab before it decays?

Short Answer

Expert verified
Answer: The muon travels on average about 15.47 x 10^6 meters in the lab frame before it decays.

Step by step solution

01

Find the time dilation factor

Use the relativistic time dilation formula to find the time dilation factor, which is given by \(\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}\), where \(v\) is the speed of the muon, and \(c\) is the speed of light. We are given \(v = 0.994c\). Plugging the values, we get: \(\gamma = \frac{1}{\sqrt{1 - \frac{(0.994c)^2}{c^2}}} = \frac{1}{\sqrt{1 - 0.994^2}}\)
02

Calculate the time dilation factor

Now, calculate the value of \(\gamma\): \(\gamma = \frac{1}{\sqrt{1 - 0.994^2}} \approx 7.089\)
03

Convert the mean lifetime to the lab frame

Using the time dilation factor, convert the mean lifetime of the muon in its rest frame to the lab frame by multiplying the rest frame mean lifetime by the time dilation factor, \(\gamma\). The mean lifetime in the rest frame, \(t_r\), is given as \(2.2 \mu s\). So, the mean lifetime in the lab frame, \(t_l\), will be: \(t_l = \gamma \cdot t_r \approx 7.089 \times 2.2 \mu s\)
04

Calculate the mean lifetime in the lab frame

Now, calculate the value of the mean lifetime in lab frame: \(t_l \approx 7.089 \times 2.2 \mu s = 15.596 \mu s\)
05

Find the average distance traveled in the lab frame

To find the average distance traveled by a muon in the lab frame before it decays, multiply the mean lifetime in the lab frame by the speed of the muon: \(distance = t_l \cdot v \approx 15.596 \mu s \times 0.994c\)
06

Calculate the average distance traveled

Finally, calculate the value of the distance: \(distance \approx 15.596 \mu s \times 0.994c \approx 15.47 \times 10^6 m\) The muon travels on average about \(15.47 \times 10^6 m\) in the lab frame before it decays.

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Most popular questions from this chapter

A lambda hyperon \(\Lambda^{0}\) (mass \(=1115.7 \mathrm{MeV} / \mathrm{c}^{2}\) ) at rest in the lab frame decays into a neutron \(n\) (mass \(=\) $939.6 \mathrm{MeV} / c^{2}\( ) and a pion (mass \)=135.0 \mathrm{MeV} / c^{2}$ ): $$\Lambda^{0} \rightarrow \mathrm{n}+\pi^{0}$$ What are the kinetic energies (in the lab frame) of the neutron and pion after the decay? [Hint: Use Eq. \((26-11)\) to find momentum.]
A futuristic train moving in a straight line with a uniform speed of \(0.80 c\) passes a series of communications towers. The spacing between the towers, according to an observer on the ground, is \(3.0 \mathrm{km} .\) A passenger on the train uses an accurate stopwatch to see how often a tower passes him. (a) What is the time interval the passenger measures between the passing of one tower and the next? (b) What is the time interval an observer on the ground measures for the train to pass from one tower to the next?
Derive the energy-momentum relation $$E^{2}=E_{0}^{2}+(p c)^{2}$$ Start by squaring the definition of total energy \(\left(E=K+E_{0}\right)\) and then use the relativistic expressions for momentum and kinetic energy $[\text{Eqs}. (26-6) \text { and }(26-8)]$.
A cosmic-ray proton entering the atmosphere from space has a kinetic energy of \(2.0 \times 10^{20} \mathrm{eV} .\) (a) What is its kinetic energy in joules? (b) If all of the kinetic energy of the proton could be harnessed to lift an object of mass \(1.0 \mathrm{kg}\) near Earth's surface, how far could the object be lifted? (c) What is the speed of the proton? Hint: \(v\) is so close to \(c\) that most calculators do not keep enough significant figures to do the required calculation, so use the binomial approximation: $$\text { if } \gamma \gg 1, \quad \sqrt{1-\frac{1}{\gamma^{2}}} \approx 1-\frac{1}{2 \gamma^{2}}$$
Kurt is measuring the speed of light in an evacuated chamber aboard a spaceship traveling with a constant velocity of \(0.60 c\) with respect to Earth. The light is moving in the direction of motion of the spaceship. Siu- Ling is on Earth watching the experiment. With what speed does the light in the vacuum chamber travel, according to Siu-Ling's observations?
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