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Chapter 37: Problem 2

The positive muon \(\left(\mu^{+}\right),\) an unstable particle, lives on average \(2.20 \times 10^{-6} \mathrm{s}\) (measured in its own frame of reference) before decaying. (a) If such a particle is moving, with respect to the laboratory, with a speed of \(0.900 c,\) what average lifetime is measured in the laboratory? (b) What average distance, measured in the laboratory, does the particle move before decaying?

Short Answer

Expert verified
(a) Lifetime is approximately 5.03 microseconds; (b) Distance is about 1.36 km.

Step by step solution

01

Understanding Time Dilation

In special relativity, time dilation is an important phenomenon. When a particle is moving at a significant fraction of the speed of light, time intervals (like the lifetime of the muon) experienced by it appear longer to an outside observer. The formula for time dilation is given by \( t = \frac{t_0}{\sqrt{1 - \left(\frac{v}{c}\right)^2}} \), where \( t_0 \) is the proper time (lifetime in the particle's frame), \( v \) is the speed, and \( c \) is the speed of light.
02

Applying Time Dilation Formula

Here, \( t_0 = 2.20 \times 10^{-6} \) seconds and \( v = 0.900c \). Plug these values into the time dilation formula: \[t = \frac{2.20 \times 10^{-6} \text{ s}}{\sqrt{1 - (0.900)^2}} \].This simplifies to \( t = \frac{2.20 \times 10^{-6}}{\sqrt{1 - 0.81}} = \frac{2.20 \times 10^{-6}}{\sqrt{0.19}} \approx 5.03 \times 10^{-6} \text{ s} \).
03

Calculating Average Distance Traveled

The average distance traveled by the particle in the laboratory can be calculated using the formula \( d = v \times t \), where \( t \) is the dilated lifetime. Knowing \( v = 0.900c \) and \( t = 5.03 \times 10^{-6} \) s, the average distance is \( d = 0.900c \times 5.03 \times 10^{-6} \text{ s} \).
04

Evaluating the Distance

Given \( c = 3.00 \times 10^8 \text{ m/s} \), substituting this into the distance formula gives:\[d = 0.900 \times 3.00 \times 10^8 \times 5.03 \times 10^{-6} = 1.36 \times 10^3 \text{ meters} \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Special Relativity
Special relativity teaches us how time and space are perceived differently for objects moving at a substantial fraction of the speed of light. This theory, proposed by Albert Einstein, describes how time can slow down for fast-moving objects, compared to those at rest. Here are some key aspects:
  • When moving close to the speed of light, time dilation occurs, causing time to appear stretched for the moving object from the perspective of a stationary observer.
  • According to the formula for time dilation, the perceived time duration, \( t \), is longer than the proper time, \( t_0 \), by a factor of \( \frac{1}{\sqrt{1 - \left(\frac{v}{c}\right)^2}} \), where \( v \) is the velocity of the object, and \( c \) is the speed of light.
Understanding these differences is crucial when we explore phenomena like muon decay.
Muon Decay
Muons are subatomic particles similar to electrons but much heavier. They are unstable and spontaneously decay after a short lifespan. Muon decay is an interesting phenomenon observed in particle physics, illustrating principle concepts of special relativity. Key points include:
  • The lifespan of a muon, called its mean lifetime, is about \( 2.20 \times 10^{-6} \) seconds in its rest frame.
  • In our exercise, the muon travels at a high velocity, \( 0.900c \), indicating that its lifetime appears longer to us due to time dilation.
This illustrates how high-velocity influences our observations of particle lifetimes.
Particle Physics
Particle physics is the branch of physics studying the smallest known particles. It provides insight into the very structure of matter and fundamental forces of nature. How muons decay is a subject within this field. Important aspects include:
  • Particles like muons are subject to decay processes where they transform into other particles.
  • Studying these decay processes helps scientists understand the fundamental forces and the basic building blocks of the universe.
Muon decay demonstrates various principles of particle physics and special relativity by showing how particles behave at high speeds.
Average Lifetime Calculation
Calculating the average lifetime of particles like muons when they move at relativistic speeds involves using special relativity principles, specifically time dilation. Here's how it works:
  • The proper lifetime of the muon in its rest frame is \( 2.20 \times 10^{-6} \) seconds.
  • To find the dilated lifetime for the laboratory observer, we apply the time dilation formula: \( t = \frac{t_0}{\sqrt{1 - (v/c)^2}} \).
  • For a muon traveling at \( 0.900c \), the calculated lifetime becomes approximately \( 5.03 \times 10^{-6} \) seconds in the lab frame.
By understanding this calculation, we comprehend how motion influences perceived time durations.

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Most popular questions from this chapter

Determining the Masses of Stars. Many of the stars in the sky are actually binary stars, in which two stars orbit about their common center of mass. If the orbital speeds of the stars are high enough, the motion of the stars can be detected by the Doppler shifts of the light they empler. Stars for which this is the case are called spectroscopic binary stars. Figure \(\mathrm{P3} 7.75\) shows the simplest case of a spectroscopic binary star: two identical stars, each with mass \(m,\) orbiting their center of mass in a circle of radius \(R\) . The plane of the stars' orbits is edge-on to the line of sight of an observer on the earth. (a) The light produced by heated hydrogen gas in a laboratory on the earth has a frequency of \(4.568110 \times\) \(10^{14}\) Hz. In the light received from the stars by a telescope on the earth, hydrogen light is observed to vary in frequency between \(4.567710 \times 10^{14} \mathrm{Hz}\) and \(4.568910 \times 10^{14} \mathrm{Hz}\) . Determine whether the binary star system as a whole is moving toward or away from the earth, the speed of this motion, and the orbital speeds of the stars. Hint: The speeds involved are much less than \(c,\) so you may use the approximate result \(\Delta f / f=u / c\) given in Section 37.6 . (b) The light from each star in the binary system varies from its maximum frequency to its minimum frequency and back again in 11.0 days. Determine the orbital radius \(R\) and the mass \(m\) of each star. Give your answer for \(m\) in kilograms and as a multiple of the mass of the sun, \(1.99 \times 10^{30}\) kg. Compare the value of \(R\) to the distance from the earth to the sun, \(1.50 \times 10^{11}\) m. (This technique is actually used in astronomy to determine the masses of stars. In practice, the problem is more complicated because the two stars in a binary system are usually not identical, the orbits are usually not circular, and the plane of the orbits is usually tilted with respect to the line of sight from the earth.)

A pursuit spacecraft from the planet Tatooine is attempting to catch up with a Trade Federation cruiser. As measured by an observer on Tatooine, the cruiser is traveling away from the planet with a speed of 0.600\(c .\) The pursuit ship is traveling at a speed of 0.800\(c\) relative to Tatooine, in the same direction as the cruiser. (a) For the pursuit ship to catch the cruiser, should the velocity of the cruiser relative to the pursuit ship be directed toward or away from the pursuit ship? (b) What is the speed of the cruiser relative to the pursuit ship?

The French physicist Armand Fizeau was the first to measure the speed of light accurately. He also found experimentally that the speed, relative to the lab frame, of light traveling in a tank of water that is itself moving at a speed \(V\) relative to the lab frame is $$v=\frac{c}{n}+k V$$ where \(n=1.333\) is the index of refraction of water. Fizeau called \(k\) the draging coefficient and obtained an experimental value of \(k=0.44 .\) What value of \(k\) do you calculate from relativistic transformations?

An unstable particle is created in the upper atmosphere from a cosmic ray and travels straight down toward the surface of the earth with a speed of 0.99540\(c\) relative to the earth. A scientist at rest on the earth's surface measures that the particle is created at an altitude of 45.0 \(\mathrm{km} .\) (a) As measured by the scientist, how much time does it take the particle to travel the 45.0 \(\mathrm{km}\) to the surface of the earth? (b) Use the length-contraction formula to calculate the distance from where the particle is created to the surface of the earth as measured in the particle's frame. (c) In the particle's frame, how much time does it take the particle to travel from where it is created to the surface of the earth? Calculate this time both by the time dilation formula and from the distance calculated in part (b). Do the two results agree?

One of the wavelengths of light emitted by hydrogen atoms under normal laboratory conditions is \(\lambda=656.3 \mathrm{nm},\) in the red portion of the electromagnetic spectrum. In the light emitted from a distant galaxy this same spectral line is observed to be Doppler- shifted to \(\lambda=953.4 \mathrm{nm},\) in the infrared portion of the spectrum. How fast are the emitting atoms moving relative to the earth? Are they approaching the earth or receding from it?
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