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Chapter 38: Problem 43

Touchstone Example \(38-7\) concluded that when a \(\mathrm{K}^{\circ}\) meson at rest decays into two daughter \(\pi\) mesons, they move in opposite directions in the rest frame of the original \(\mathrm{K}^{\circ}\) meson, each with a speed of \(0.828 c\). Now suppose that the initial \(\mathrm{K}^{\circ}\) meson moves with speed \(v^{\text {rel }}=0.9 c\) as measured in the laboratory frame. What are the maximum and minimum speeds of the daughter \(\pi\) mesons with respect to the laboratory?

Short Answer

Expert verified
The maximum speed is \(0.987 c\), and the minimum speed is \(0.351 c\).

Step by step solution

01

- Define the reference frames and given speeds

Identify the rest frame of the \(\text{K}^{\circ}\) meson and the laboratory frame. In the rest frame of the \(\text{K}^{\circ}\) meson, the daughter \(\text{\pi}\) mesons move with a speed of \(v = 0.828 c\). The \(\text{K}^{\circ}\) meson moves with a speed of \(0.9 c\) in the laboratory frame.
02

- Use velocity addition formula

Use the relativistic velocity addition formula to find the speeds of the \(\text{\pi}\) mesons in the laboratory frame. The velocity addition formula is \[v' = \frac{v + u}{1 + \frac{vu}{c^2}}\] where \(v\) is the speed of the \(\text{\pi}\) meson in the rest frame of the \(\text{K}^{\circ}\) meson, and \(u\) is the speed of the \(\text{K}^{\circ}\) meson relative to the laboratory frame.
03

- Calculate the maximum speed

For the \(\text{\pi}\) meson moving in the same direction as the \(\text{K}^{\circ}\) meson, substitute \(v = 0.828 c\) and \(u = 0.9 c\) into the velocity addition formula: \[v'_{\text{max}} = \frac{0.828 c + 0.9 c}{1 + \frac{(0.828)(0.9)}{1}} = 0.987 c\]
04

- Calculate the minimum speed

For the \(\text{\pi}\) meson moving opposite to the direction of the \(\text{K}^{\circ}\) meson, substitute \(v = -0.828 c\) and \(u = 0.9 c\) into the velocity addition formula: \[v'_{\text{min}} = \frac{-0.828 c + 0.9 c}{1 + \frac{(-0.828)(0.9)}{1}} = 0.351 c\]

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Key Concepts

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

K meson decay
The K meson, also known as the K0 meson or kaon, is an unstable subatomic particle. When it decays, it typically breaks down into lighter particles. In our exercise, the K0 meson decays into two pi mesons (蟺 mesons). This process is called 'decay' because the original particle doesn't last long and transforms into other particles.
pi mesons
Pi mesons, or pions, are subatomic particles involved in holding the atomic nucleus together. They come in different charges: 蟺+, 蟺-, and 蟺0. In our example, when the K0 meson decays at rest, the resulting pi mesons move in opposite directions. This ensures the conservation of momentum, a key principle in physics.
velocity addition formula
In special relativity, the velocities of moving bodies don't simply add up like they do in everyday experience. Instead, we use the relativistic velocity addition formula:
The formula states:
For the maximum speed calculation:
For the 蟺 meson moving in the same direction as the K0 meson, substitute velocity values
For the minimum speed calculation:
For the 蟺 meson moving in the opposite direction as the K0 meson, substitute velocity values
naturally different directions depending on the task
special relativity
Special relativity is a fundamental theory in physics, developed by Albert Einstein. It's crucial when dealing with objects moving at speeds close to the speed of light, denoted as 'c'. Two core ideas in special relativity are: 1. The laws of physics are the same in all inertial frames.
2. The speed of light is constant, regardless of the observer's frame of reference.

These principles lead to the relativistic velocity addition formula, which ensures that no object's speed exceeds the speed of light, redefining how velocities combine.

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Most popular questions from this chapter

Through what voltage must an electron be accelerated from rest in order to increase its energy to \(101 \%\) of its rest energy?

You wish to make \(a\) round trip from Earth in a spaceship, traveling at constant speed in a straight line for 6 months on your watch and then returning at the same constant speed. You wish, further, to find Earth to be 1000 years older on your return. (a) What is the value of your constant speed with respect to Earth? (b) How much do you age during the trip? (c) Does it matter whether or not you travel in a straight line? For example, could you travel in a huge circle that loops back to Earth?

A particle moves with uniform speed \(v_{y}^{\prime}=\Delta y^{\prime} / \Delta t^{\prime}\) in the \(y^{\prime}\) direction with respect to a rocket frame that moves along the \(x\) axis of a laboratory frame. Find exprcssions for the \(x\) -componcnt and for the \(y\) -componcnt of the particle's velocity in the laboratory frame.

A gamma ray (an energetic photon) falls on a nucleus of initial mass \(m\), initially at rest. The energy \(E_{\mathrm{p}}\) of the incoming gamma ray matches the energy separation between the lowest energy of the nucleus and its first excited state, so the incident photon is absorbed. We want to know the mass \(m^{*}\) of the excited nucleus. (see Fig. \(38-13 .\) ) (a) Show that the conservation of energy and momentum equations are, in an obvious notation: and $$ \begin{array}{c} E_{\mathrm{p}}+m c^{2}=E_{m^{*}} \\ \frac{E_{\mathrm{p}}}{c}=p_{m^{*}}=\frac{\left(E_{m^{*}}^{2}-m^{* 2} c^{4}\right)^{1 / 2}}{c} . \end{array} $$ (b) Combine the two conservation equations to find an expression for \(m^{*}\) as a function of \(E_{\mathrm{p}}, m\), and \(c\). (c) Show that for very small values of \(E_{\mathrm{p}}\) the limiting result is \(m^{*}=m .\) Explain why this limiting result is reasonable.

In the 24 th century the fastest available interstellar rocket moves at \(v=0.75 c .\) Mya Allen is sent in this rocket at full (constant) speed to Sirius, the Dog Star, the brightest star in the heavens as seen from Earth, which is a distance \(8.7\) ly as measured in the Earth frame. Assume Sirius is at rest with respect to Earth. Mya stays near Sirius, slowly orbiting around that Dog Star, for 7 years as recorded on her wristwatch while making observations and recording data, then returns to Earth with the same speed \(v=0.75 \mathrm{c}\). According to Earth-linked observers: (a) When does Mya arrive at Sirius? (b) When does Mya leave Sirius? (c) When does Mya arrive back at Earth? According to Mya's wristwatch: (d) When does she arrive at Sirius? (e) When does she leave Sirius? (f) When does she arrive back on Earth?
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