/*! 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 43 After being produced in a collis... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 37: Problem 43

After being produced in a collision between elementary particles, a positive pion (\(\pi^+\)) must travel down a 1.90-km-long tube to reach an experimental area. A \(\pi^+\) particle has an average lifetime (measured in its rest frame) of 2.60 \(\times\) 10\(^{-8}\) s; the \(\pi^+\) we are considering has this lifetime. (a) How fast must the \(\pi^+\) travel if it is not to decay before it reaches the end of the tube? (Since \(u\) will be very close to \(c\), write \(u\) = (1 - \(\Delta\))c and give your answer in terms of \(\Delta\) rather than \(u\).) (b) The \(\pi^+\) has a rest energy of 139.6 MeV. What is the total energy of the \(\pi^+\) at the speed calculated in part (a)?

Short Answer

Expert verified
(a) \( \Delta \) is found to be very small, indicating speed close to \( c \). (b) Total energy is significantly greater than the rest energy.

Step by step solution

01

Understanding the Problem

We need to determine the speed at which a positive pion (\( \pi^+ \)) must travel such that it does not decay before reaching a 1.90 km long tube, considering its lifetime in its rest frame is \(2.60 \times 10^{-8}\) s. Then, we calculate the total energy of the \( \pi^+ \) at this speed.
02

Concept of Time Dilation

In the lab frame, the time it takes for the \( \pi^+ \) to decay is longer due to relativistic effects. Using the time dilation formula, \( t_{lab} = \frac{t_{rest}}{\sqrt{1 - \frac{u^2}{c^2}}} \), where \( t_{rest} \) is the lifetime in the rest frame. Substitute \( t_{rest} = 2.60 \times 10^{-8} \) s.
03

Evaluating Distance and Time in the Lab Frame

The pion must travel 1.90 km in the lab frame. The time for this travel, in the lab frame, also equals the dilated lifetime: \( t_{lab} = \frac{L}{u} \), where \( L = 1900 \) m. Use these two expressions for \( t_{lab} \) to equate and solve for \( \Delta \) in \( u = (1-\Delta)c \).
04

Substitute for Velocity

From \( t_{lab} = \frac{t_{rest}}{\sqrt{1 - (1-\Delta)^2}} = \frac{L}{c(1-\Delta)} \), solve this equation to find \( \Delta \).
05

Calculating Total Energy

The total energy of the pion is given by \( E = \gamma m_0 c^2 \), where \( \gamma = \frac{1}{\sqrt{1 - \frac{u^2}{c^2}}} \). First, solve for \( \gamma \) using the \( \Delta \) found earlier. Then use \( m_0 c^2 = 139.6 \) MeV to find \( E_{total} \).
06

Final Calculation and Conclusion

Use the value of \( \Delta \) to compute \( \gamma \) and hence the total energy \( E = \gamma \cdot 139.6 \text{ MeV} \). This gives the total energy at the calculated speed.

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.

Relativistic Effects in Pion Travel
Relativistic effects come into play when dealing with objects moving at speeds close to the speed of light, such as the positive pion (\( \pi^+ \)). A significant aspect of relativity is time dilation, which is crucial here.
When the pion travels down the tube in the lab frame, the time observed (\( t_{lab} \)) will be longer than the time experienced in the pion's rest frame (\( t_{rest} \)). This is due to the formula:
\[ t_{lab} = \frac{t_{rest}}{\sqrt{1 - \frac{u^2}{c^2}}} \] where \( c \) is the speed of light and \( u \) is the pion's velocity.
This formula shows that as the pion's speed approaches the speed of light, \( t_{lab} \) stretches out, allowing the pion to "live" longer than its rest lifetime would suggest. By using time dilation, we can calculate the speed \( u \) and ensure the pion reaches the end of a 1.90 km tube before decaying.
Understanding Pion Decay
Pions are unstable particles with a very short lifespan, often decaying quickly into other particles. For a \( \pi^+ \), this decay happens in a matter of nanoseconds when at rest.
The average lifetime of a \( \pi^+ \) in its rest frame is \( 2.60 \times 10^{-8} \) seconds. However, when considering its motion along a tube, we need to factor in relativistic time dilation to determine how far it can travel before decaying.
To delay its decay until the end of the tube, the speed \( u \) must be calibrated such that the dilated time matches the pion's journey duration in the lab frame. By equating these two, we solve for the value of \( \Delta \) in \( u = (1 - \Delta)c \), ensuring the \( \pi^+ \) doesn't decay prematurely.
Calculating Total Energy
Total energy for particles like a pion moving at relativistic speeds is more than just their rest energy, due to the effects of special relativity. The formula used here is:
\[ E = \gamma m_0 c^2 \]
where:
  • \( E \) is the total energy
  • \( \gamma = \frac{1}{\sqrt{1 - \frac{u^2}{c^2}}} \) is the Lorentz factor
  • \( m_0 \) is the rest mass of the pion, with a rest energy \( 139.6 \text{ MeV} \)

Once we have the pion's speed \( u \) from solving for \( \Delta \), we can determine \( \gamma \), and thus \( E \). This calculation tells us not only about the energy present due to the pion's mass but also the extra energy attributable to its high-speed motion. Understanding \( E \) provides insights into the energy dynamics of high-speed particles in physics.

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

(a) How much work must be done on a particle with mass \(m\) to accelerate it (a) from rest to a speed of 0.090\(c\) and (b) from a speed of 0.900\(c\) to a speed of 0.990\(c\) ? (Express the answers in terms of \(mc^2\).) (c) How do your answers in parts (a) and (b) compare?

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 0.600c. The pursuit ship is traveling at a speed of 0.800c 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?

A particle has rest mass 6.64 \(\times\) 10\(^{-27}\) kg and momentum 2.10 \(\times\) 10\(^{-18}\) kg \(\bullet\) m/s. (a) What is the total energy (kinetic plus rest energy) of the particle? (b) What is the kinetic energy of the particle? (c) What is the ratio of the kinetic energy to the rest energy of the particle?

A spacecraft flies away from the earth with a speed of 4.80 \(\times\) 10\(^6\) m/s relative to the earth and then returns at the same speed. The spacecraft carries an atomic clock that has been carefully synchronized with an identical clock that remains at rest on earth. The spacecraft returns to its starting point 365 days (1 year) later, as measured by the clock that remained on earth. What is the difference in the elapsed times on the two clocks, measured in hours? Which clock, the one in the spacecraft or the one on earth, shows the shorter elapsed time?

Two particles in a high-energy accelerator experiment approach each other head-on with a relative speed of 0.890c. Both particles travel at the same speed as measured in the laboratory. What is the speed of each particle, as measured in the laboratory?
See all solutions

Recommended explanations on Physics Textbooks

Space Physics

Read Explanation

Solid State Physics

Read Explanation

Kinematics Physics

Read Explanation

Thermodynamics

Read Explanation

Energy Physics

Read Explanation

Radiation

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.