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Chapter 11: Problem 81

Plans for an accelerator that produces a secondary beam of K mesons to scatter from nuclei, for the purpose of studying the strong force, call for them to have a kinetic energy of \(500 \mathrm{MeV} .\) (a) What would the relativistic quantity \(\gamma=\frac{1}{\sqrt{1-v^{2} / c^{2}}}\) be for these particles? (b) How long would their average lifetime be in the laboratory? (c) How far could they travel in this time?

Short Answer

Expert verified
The relativistic quantity γ for K mesons with a kinetic energy of 500 MeV is approximately 2.02. Their average lifetime in the laboratory is about \(2.5048 × 10^{-8} \text{s}\). In this time, they could travel approximately 42 meters.

Step by step solution

01

Find the velocity using the kinetic energy formula

We know the kinetic energy of a particle is given by the formula: \[ KE = (\gamma - 1)m_0c^2 \] Here, KE is the kinetic energy, γ is the relativistic factor, \(m_0\) is the rest mass, and c is the speed of light. We are given the kinetic energy (KE) as 500 MeV. To convert it to Joules, we can use the following relation: 1 eV = 1.6 × 10^-19 J. The rest mass of a K meson is about \(m_0 = 494 \text{ MeV}/c^2\). First, convert the kinetic energy to Joules: \[ KE = 500 \text{ MeV} × 1.6 × 10^{-13} \text{J/MeV} = 8×10^{-11} \text{J} \] Now we can rearrange the formula to solve for γ: \[ \gamma = \frac{KE}{m_0c^2} + 1 \]
02

Calculate γ

Plug in the given values to find γ: \[ \gamma = \frac{8×10^{-11} \text{ J}}{(494 \text{ MeV}) × 1.6 × 10^{-13} \text{J/MeV} × c^2} + 1 \] \[ \gamma \approx 2.02 \]
03

Find the average lifetime of K mesons in the laboratory

Given the average lifetime of K mesons at rest is \(\tau_0 = 1.24×10^{-8} \text{s}\), we can find the average lifetime in the laboratory using the time dilation formula: \[ \tau = \gamma \tau_0 \] \[ \tau = 2.02 × 1.24 × 10^{-8} \text{s} \] \[ \tau \approx 2.5048 × 10^{-8} \text{s} \]
04

Calculate the travel distance

To find the distance that the K mesons travel in the average lifetime in the laboratory, we can multiply their average velocity by their average lifetime. The average velocity can be calculated using the following formula: \[v = c \sqrt{1 - \frac{1}{\gamma^2}} \] \[ v = c\sqrt{1 - \frac{1}{ (2.02)^2}} \] Calculate the velocity: \[ v \approx 0.558c \] Now calculate the distance traveled in the average lifetime in the laboratory: \[ d = v \tau \] \[ d = 0.558c × 2.5048 × 10^{-8} \text{s} \] Here, c = 3 × 10^8 m/s , plug this value: \[ d \approx 0.558 × 3 × 10^8 \text{ m/s} × 2.5048 × 10^{-8} \text{s} \] \[ d \approx 42 \text{ m} \] So, the K mesons could travel approximately 42 meters in the given time.

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Most popular questions from this chapter

Express the \(\beta\) decays \(n \rightarrow p+e^{-}+\bar{\nu} \quad\) and \(p \rightarrow n+e^{+}+\nu\) in terms of \(\beta\) decays of quarks. Check to see that the conservation laws for charge, lepton number, and baryon number are satisfied by the quark \(\beta\) decays.

Which of the following reactions cannot because the law of conservation of strangeness is violated? (a) \(\mathrm{p}+\mathrm{n} \rightarrow \mathrm{p}+\mathrm{p}+\pi^{-}\) (b) \(\mathrm{p}+\mathrm{n} \rightarrow \mathrm{p}+\mathrm{p}+\mathrm{K}^{-}\) (c) \(\mathrm{K}^{-}+\mathrm{p} \rightarrow \mathrm{K}^{-}+\sum^{+}\) (d) \(\pi^{-}+\mathrm{p} \rightarrow \mathrm{K}^{+}+\sum^{-}\) (e) \(\mathrm{K}^{-}+\mathrm{p} \rightarrow \Xi^{0}+\mathrm{K}^{+}+\pi^{-}\) (f) \(\mathrm{K}^{-}+\mathrm{p} \rightarrow \Xi^{0}+\pi^{-}+\pi^{-}\) (g) \(\pi^{+}+\mathrm{p} \rightarrow \Sigma^{+}+\mathrm{K}^{+}\) (h) \(\pi^{-}+\mathrm{n} \rightarrow \mathrm{K}^{-}+\Lambda^{0}\)

What is the general quark composition of a baryon? Of a meson?

Each of the following strong nuclear reactions is forbidden. Identify a conservation law that is violated for each one. (a) \(\mathrm{p}+\overline{\mathrm{p}} \rightarrow \mathrm{p}+\mathrm{n}+\overline{\mathrm{p}}\) (b) \(\mathrm{p}+\mathrm{n} \rightarrow \mathrm{p}+\overrightarrow{\mathrm{p}}+\mathrm{n}+\pi^{+}\) (c) \(\pi^{-}+\mathrm{p} \rightarrow \Sigma^{+}+\mathrm{K}^{-}\) (d) \(\mathrm{K}^{-}+\mathrm{p} \rightarrow \Lambda^{0}+\mathrm{n}\)

Using the Heisenberg uncertainly principle, determine the range of the weak force if this force is produced by the exchange of a Z boson.
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