Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 20: Problem 4

The mean lifetime of muons stopped in a lead block in the laboratory is measured to be \(2.20 \mu \mathrm{s}\). The mean lifetime of high-speed muons in a burst of cosmic rays observed from the Earth is measured to be \(16.0 \mu \mathrm{s}\). Find the speed of these cosmic ray muons.

Short Answer

Expert verified
The speed of the cosmic ray muons is approximately \(2.985 × 10^8 \mathrm{m/s}\), which is very close to the speed of light.

Step by step solution

01

Identify given values

From the problem, we know that the laboratory time of the muon, \(t_0\), is \(2.20 \mu \mathrm{s}\) and the observed time, \(t\), of the cosmic rays is \(16.0 \mu \mathrm{s}\). The speed of light, \(c\), is a known constant that is approximately \(3.00 × 10^8 \mathrm{m/s}\).
02

Rearrange the time dilation formula to solve for velocity, \(v\)

We need to isolate \(v\) in the equation, which gives us the relationship: \(v = c*\sqrt{1-(t_0/t)^2}\).
03

Substitute the known values into the equation

Now we substitute \(t_0 = 2.20 \mu \mathrm{s}\), \(t = 16.0 \mu \mathrm{s}\), and \(c = 3.00 × 10^8 \mathrm{m/s}\) into the equation to get the velocity of the cosmic ray muons.
04

Compute the solution

By plugging the values into the equation, we solve for the velocity, which is very nearly the speed of light.

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

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

Find the speed parameter of a particle that takes 2 years longer than light to travel a distance of \(6.0\) ly.

Quite apart from effects due to the Earth's rotational and orbital motions, a laboratory frame is not strictly an inertial frame because a particle placed at rest there will not, in general, remain at rest; it will fall under gravity. Often, however, events happen so quickly that we can ignore free fall and treat the frame as inertial. Consider, for example, a \(1.0-\mathrm{MeV}\) electron (for which \(v=0.941 c\) ) projected horizontally into a laboratory test chamber and moving through a distance of \(20 \mathrm{~cm}\). (a) How long would it take, and \((b)\) how far would the electron fall during this interval? What can you conclude about the suitability of the laboratory as an inertial frame in this case?

A spaceship of rest length \(130 \mathrm{~m}\) drifts past a timing station at a speed of \(0.740 c .(a)\) What is the length of the spaceship as measured by the timing station? \((b)\) What time interval between the passage of the front and back end of the ship will the station monitor record?

Find the momentum of a particle of mass \(m\) in order that its total energy be three times its rest energy.

Find the speed parameter \(\beta\) and the Lorentz factor \(\gamma\) for a particle whose kinetic energy is \(10 \mathrm{MeV}\) if the particle is \((a)\) an electron, \((b)\) a proton, and \((c)\) an alpha particle.
See all solutions

Recommended explanations on Physics Textbooks

Kinematics Physics

Read Explanation

Magnetism

Read Explanation

Magnetism and Electromagnetic Induction

Read Explanation

Fields in Physics

Read Explanation

Atoms and Radioactivity

Read Explanation

Quantum Physics

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.

Sign-up for free