/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 5 The negative pion \(\left(\pi^{-... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 37: Problem 5

The negative pion \(\left(\pi^{-}\right)\) is an unstable particle with an average lifetime of \(2.60 \times 10^{-8} \mathrm{~s}\) (measured in the rest frame of the pion). (a) If the pion is made to travel at very high speed relative to a laboratory, its average lifetime is measured in the laboratory to be \(4.20 \times 10^{-7} \mathrm{~s}\). Calculate the speed of the pion expressed as a fraction of \(c\). (b) What distance, measured in the laboratory, does the pion travel during its average lifetime?

Short Answer

Expert verified
The speed of the pion as a fraction of the speed of light, c, is approximately \(0.9923c\). The distance travelled by the pion during its average lifetime in the laboratory frame is approximately 125.54 meters.

Step by step solution

01

Calculating the speed of the pion (Part a)

Using the time dilation formula and solving for \(v/c\), we get \(v/c = \sqrt{1 - (T_0/T)^2}\) where T= \(4.20 \times 10^{-7} \mathrm{s}\) and \(T_0 = 2.60 \times 10^{-8} \mathrm{s}\). Plugging these values in yields \(v/c = \sqrt{1 - (2.60 \times 10^{-8} / 4.20 \times 10^{-7})^2}\).
02

Getting the numerical answer (Part a)

Evaluating the above expression with a calculator gives a numerical result of \(v/c = 0.9923\), which is the fraction of the speed of light at which the pion travels.
03

Calculating the distance travelled by the pion (Part b)

The distance travelled by the pion during its average lifetime can be calculated as D = vT. Substituting the values for v and T, we get \(D = c \times 0.9923 \times 4.20 \times 10^{-7}\) s.
04

Getting the numerical answer (Part b)

Evaluating the above expression gives a numerical result of \(D = 125.54 m\), which is the distance travelled by the pion in the laboratory frame during its average lifetime.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Special Relativity
Special Relativity is a fundamental theory in physics, introduced by Albert Einstein in 1905. It revolutionized our understanding of space, time, and motion. The key idea is that the laws of physics are the same for all observers, no matter how fast they are moving relative to one another. A cornerstone of this theory is
  • the principle of the constancy of the speed of light - meaning that the speed of light is always the same, regardless of the motion of the light source or observer.
  • Time dilation, a bizarre but real consequence of this, results in different measurements of time for two observers moving relative to each other. To an observer in motion, time appears to move slower compared to someone at rest.
This means when dealing with particles moving at speeds close to the speed of light, such as a pion, traditional physics doesn't always hold and we must apply the principles of special relativity.
Particle Physics
Particle Physics is the study of the fundamental constituents of matter and the forces governing their interactions. Pions are a type of meson, which are intermediary particles playing a significant role in particle physics, especially in the strong force, which holds the nucleus of an atom together.
  • Pions are composed of a quark and an antiquark. They are crucial in mediating strong interactions between nucleons (protons and neutrons).
  • Though they are not fundamental particles like electrons but are still integral to the field due to their role in nuclear forces.
  • In experiments, pions are often observed near the speed of light, allowing physicists to explore the principles of special relativity in high-energy environments.
This makes them a perfect subject to study the implications of relativistic physics.
Speed of Light
The speed of light in a vacuum, denoted as \(c\), is approximately \(3.00 \times 10^8\) meters per second. This value is not just a common speed limit but a universal constant in the realm of physics. It plays a pivotal role in the theory of relativity.
  • In the context of time dilation, when particles such as pions travel at a speed close to \(c\), time appears to slow down for them relative to a stationary observer.
  • The formula \(v/c = \sqrt{1 - (T_0/T)^2}\) provides us with the means to calculate a particle's speed as a fraction of the speed of light, \(c\), enabling us to understand how special relativity impacts measurements of space and time for fast-moving objects.
  • Because nothing with mass can break this speed limit, observing particles traveling near this maximum offers valuable insights into the truly relativistic nature of the cosmos.
Pion Decay
Pion decay is a process that demonstrates some of the fascinating phenomena in particle physics. A pion is an unstable particle, typically existing for a mere fraction of a second.
  • The characteristic decay of a pion into other particles (usually muons and neutrinos) allows researchers to gather information on the properties and interactions of subatomic particles.
  • The average lifetime of a pion, about \(2.60 \times 10^{-8}\) seconds in its own rest frame, extends significantly when it travels at near-light speeds, due to time dilation.
  • This not only highlights the effects of relativistic speeds on decay rates but also supports the predictions of special relativity, providing real-world evidence of its validity.
Relativistic Physics
Relativistic Physics encompasses the study of systems where relativistic speeds are involved, often close to the speed of light. It profoundly affects how we understand space and time.
  • Relativistic effects become noticeable in fast-moving particles, such as pions in particle accelerators, where their measured lifetimes and distances traveled differ from non-relativistic predictions.
  • The conversion of mass into energy, as described by Einstein's famous equation \(E=mc^2\), is a vital part of relativistic physics, demonstrating the interconnectedness of energy and matter.
  • Observations of high-speed particles allow scientists to test and confirm the principles of relativity, offering a glimpse into the universe's extreme environments, like those near black holes or during the Big Bang.
Understanding relativistic physics is key to unraveling the mysteries of the cosmos and the fundamental interactions at the smallest scales.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Electrons are accelerated through a potential difference of \(750 \mathrm{kV},\) so that their kinetic energy is \(7.50 \times 10^{5} \mathrm{eV}\). (a) What is the ratio of the speed \(v\) of an electron having this energy to the speed of light, \(c ?\) (b) What would the speed be if it were computed from the principles of classical mechanics?

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 emit. Stars for which this is the case are called spectroscopic binary stars. Figure \(\mathbf{P 3 7 . 6 8}\) 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 \(4.568110 \times 10^{14} \mathrm{~Hz}\) 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, \(1.99 \times 10^{30} \mathrm{~kg} .\) Compare the value of \(R\) to the distance from the earth to the sun, \(1.50 \times 10^{11} \mathrm{~m}\). (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 source of electromagnetic radiation is moving in a radial direction relative to you. The frequency you measure is 1.25 times the frequency measured in the rest frame of the source. What is the speed of the source relative to you? Is the source moving toward you or away from you?

(a) Through what potential difference does an electron have to be accelerated, starting from rest, to achieve a speed of \(0.980 c ?\) (b) What is the kinetic energy of the electron at this speed? Express your answer in joules and in electron volts.

Muons are unstable subatomic particles that decay to electrons with a mean lifetime of \(2.2 \mu \mathrm{s}\) They are produced when cosmic rays bombard the upper atmosphere about \(10 \mathrm{~km}\) above the earth's surface, and they travel very close to the speed of light. The problem we want to address is why we see any of them at the earth's surface. (a) What is the greatest distance a muon could travel during its \(2.2 \mu\) s lifetime? (b) According to your answer in part (a), it would seem that muons could never make it to the ground. But the \(2.2 \mu \mathrm{s}\) lifetime is measured in the frame of the muon, and muons are moving very fast. At a speed of \(0.999 c,\) what is the mean lifetime of a muon as measured by an observer at rest on the earth? How far would the muon travel in this time? Does this result explain why we find muons in cosmic rays? (c) From the point of view of the muon, it still lives for only \(2.2 \mu \mathrm{s},\) so how does it make it to the ground? What is the thickness of the \(10 \mathrm{~km}\) of atmosphere through which the muon must travel, as measured by the muon? Is it now clear how the muon is able to reach the ground?
See all solutions

Recommended explanations on Physics Textbooks

Kinematics Physics

Read Explanation

Famous Physicists

Read Explanation

Work Energy and Power

Read Explanation

Electricity and Magnetism

Read Explanation

Physical Quantities And Units

Read Explanation

Linear Momentum

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.