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The positive muon (\(\mu^+\)), an unstable particle, lives on average 2.20 \(\times\) 10\(^{-6}\) 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 0.900c, 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) Laboratory lifetime is 5.05 \( \times \) 10\(^{-6}\) seconds. (b) Distance traveled is 1.36 \( \times \) 10\(^3\) meters.

Step by step solution

01

Understand Time Dilation

The lifetime of a moving particle as measured in the laboratory frame differs from the lifetime measured in its own frame due to time dilation, a consequence of Einstein’s theory of relativity. The time dilation formula is \( t = \frac{t_0}{\sqrt{1 - \left(\frac{v}{c}\right)^2}} \), where \( t_0 \) is the proper lifetime, \( v \) is the speed of the particle, and \( c \) is the speed of light.
02

Calculate the Dilated Lifetime

Using the time dilation formula, substitute \( t_0 = 2.20 \times 10^{-6} \) s, \( v = 0.900c \), and \( c \) as the speed of light. The laboratory lifetime \( t \) is given by \[ t = \frac{2.20 \times 10^{-6}}{\sqrt{1 - (0.900)^2}}. \]Calculate the square root term and complete the division to find \( t \).
03

Plug in Values and Simplify

Calculate \( 1 - (0.900)^2 = 1 - 0.81 = 0.19 \). Consequently, \( \sqrt{0.19} \approx 0.436 \). Substitute this result back into the equation: \[ t = \frac{2.20 \times 10^{-6}}{0.436}. \]Divide to find \( t \approx 5.05 \times 10^{-6} \) seconds.
04

Calculate the Distance Traveled

In the laboratory frame, the distance \( d \) the particle travels is given by \( d = v \times t \). Substitute \( v = 0.900c \) and \( t = 5.05 \times 10^{-6} \) s. The distance \( d \) is \[ d = 0.900 \times 3 \times 10^8 \times 5.05 \times 10^{-6}. \]Calculate this multiplication to find \( d \).
05

Final Calculation and Solution

Calculate \( d = 0.900 \times 3 \times 10^8 \times 5.05 \times 10^{-6} = 1.36 \times 10^3 \) meters. This is the average distance the muon travels before decaying as measured in the laboratory.

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Key Concepts

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

Relativistic Effects
Relativistic effects are fascinating phenomena that arise due to observing events at speeds approaching the speed of light. One such effect is time dilation, which plays a pivotal role in modern physics. When particles like muons travel at velocities close to the speed of light, their behaviors contrast significantly with classical expectations. This is because relativistic effects cause time to stretch from the perspective of an external observer. A moving clock ticks slower compared to a stationary clock due to this phenomenon.
  • Objects moving near light speed experience changes in mass, time, and length.
  • Time dilation is a key factor in understanding these changes.
  • The formula for time dilation involves the speed of the particle relative to the speed of light.
Time dilation allows us to calculate how long a particle's lifetime appears from the lab compared to being at rest, which is foundational in understanding high-speed particle behaviors.
Particle Lifetime
The lifespan of particles, such as muons, greatly impacts their study in physics. Measured in their rest frame, the lifetime of a positive muon is an average of 2.20 microseconds. However, this lifetime increases when observed from a stationary frame because of time dilation.
  • At rest, the muon's average lifetime is exactly 2.20 microseconds.
  • When moving, the lifetime is perceived as longer by observers.
  • This dilation is a result of the particle's speed approaching light speed.
Understanding particle lifetime requires recognizing that these measurements differ based on the observer’s frame of reference. This change in perception helps scientists predict how far particles can travel before decaying, aiding in experimental physics.
Muon Decay
Muon decay is a process in which a muon, an elementary particle similar to the electron, transforms into other particles. This typically involves the conversion into an electron, a neutrino, and an antineutrino. The decay process helps physicists understand particle interactions and the conservation laws in particle physics.
  • Muons are heavier cousins of electrons and unstable.
  • Understanding their decay properties lets physicists explore fundamental forces.
  • Measuring travel distance and decay helps in exploring relativistic effects.
The study of muon decay processes is critical in experimental physics, particularly in understanding how conditions like near-light speeds influence particle behavior and decay pathways.
Einstein's Theory of Relativity
Einstein's theory of relativity is a groundbreaking concept that revolutionized our understanding of space, time, and energy. It comprises the special and general theories of relativity, with the former particularly pertinent to inertial frames of reference and the latter to gravitational fields.
  • Special relativity describes how time and space are interconnected.
  • General relativity extends these concepts to include gravity's influence.
  • Special relativity predicts time dilation and mass increase at high speeds.
The theory provides the framework for analyzing particle dynamics at high speeds. For example, Einstein’s equations help us understand how particles like muons behave when moving close to the speed of light, explaining why they appear to live longer than they do at rest. This theory underscores nearly all modern physics pertaining to fast-moving objects and their observed behaviors.

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Most popular questions from this chapter

Two protons (each with rest mass \(M\) = 1.67 \(\times\) 10\(^{-27}\) kg) are initially moving with equal speeds in opposite directions. The protons continue to exist after a collision that also produces an \(\eta_0\) particle (see Chapter 44). The rest mass of the \(\eta_0\) is m = 9.75 \(\times\) 10\(^{-28}\) kg. (a) If the two protons and the \(\eta_0\) are all at rest after the collision, find the initial speed of the protons, expressed as a fraction of the speed of light. (b) What is the kinetic energy of each proton? Express your answer in MeV. (c) What is the rest energy of the \(\eta_0\), expressed in MeV? (d) Discuss the relationship between the answers to parts (b) and (c). 37.39 .

The negative pion (\(\pi^-\)) is an unstable particle with an average lifetime of 2.60 \(\times\) 10\(^{-8}\)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\(^{-7}\) 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?

A cube of metal with sides of length \(a\) sits at rest in a frame \(S\) with one edge parallel to the \(x\)-axis. Therefore, in \(S\) the cube has volume \(a^3\). Frame \(S'\) moves along the \(x\)-axis with a speed \(u\). As measured by an observer in frame \(S'\), what is the volume of the metal cube?

A spaceship moving at constant speed u relative to us broadcasts a radio signal at constant frequency \(f_0\) . As the spaceship approaches us, we receive a higher frequency \(f\) ; after it has passed, we receive a lower frequency. (a) As the spaceship passes by, so it is instantaneously moving neither toward nor away from us, show that the frequency we receive is not \(f_0\) , and derive an expression for the frequency we do receive. Is the frequency we receive higher or lower than \(f_0\)? (\(Hint\): In this case, successive wave crests move the same distance to the observer and so they have the same transit time. Thus \(f\) equals 1 /T. Use the time dilation formula to relate the periods in the stationary and moving frames.) (b) A spaceship emits electromagnetic waves of frequency \(f_0\) = 345 MHz as measured in a frame moving with the ship. The spaceship is moving at a constant speed 0.758\(c\) relative to us. What frequency \(f\) do we receive when the spaceship is approaching us? When it is moving away? In each case what is the shift in frequency, \(f - f_0\)? (c) Use the result of part (a) to calculate the frequency \(f\) and the frequency shift (\(f - f_0\)) we receive at the instant that the ship passes by us. How does the shift in frequency calculated here compare to the shifts calculated in part (b)?

Electromagnetic radiation from a star is observed with an earth-based telescope. The star is moving away from the earth at a speed of 0.520c. If the radiation has a frequency of 8.64 \(\times\) 10\(^{14}\) Hz in the rest frame of the star, what is the frequency measured by an observer on earth?

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