University Physics with Modern Physics

Roger A. Freedman, Hugh D. Young

Chapter 44

Particle Physics and cosmology - all with Video Answers

Educators

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Chapter Questions

Problem 1

The starship Enterprise, of television and movie fame, is powered by combining matter and antimatter. If the entire $400-\mathrm{kg}$ antimatter fuel supply of the Enterprise combines with matter, how much energy is released? How does this compare to the U.S. yearly energy use, which is roughly $1.0 \times 10^{20} \mathrm{~J}$ ?

Joshua Young

Joshua Young

Numerade Educator

Problem 2

Two equal-energy photons collide head-on and annihilate each other, producing a $\mu^{+} \mu^{-}$ pair. The muon mass is given in terms of the electron mass in Section $44.1 .$ (a) Calculate the maximum wave- length of the photons for this to occur. If the photons have this wavelength, describe the motion of the $\mu^{+}$ and $\mu^{-}$ immediately after they are produced. (b) If the wavelength of each photon is half the value calculated in part (a), what is the speed of each muon after they have moved apart? Use correct relativistic expressions for momentum and energy.

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Erika Lynn

Numerade Educator

Problem 3

A positive pion at rest decays into a positive muon and a neutrino. (a) Approximately how much energy is released in the decay? (Assume the neutrino has zero rest mass. Use the muon and pion masses given in terms of the electron mass in Section $44.1 .$ ) (b) Why can't a positive muon decay into a positive pion?

Joshua Young

Numerade Educator

Problem 4

A proton and an antiproton annihilate, producing two photons. Find the energy, frequency, and wavelength of each photon (a) if the $\mathrm{p}$ and $\overline{\mathrm{p}}$ are initially at rest and $(\mathrm{b})$ if the $\mathrm{p}$ and $\overline{\mathrm{p}}$ collide head-on, each with an initial kinetic energy of $620 \mathrm{MeV}$.

Joshua Young

Numerade Educator

Problem 5

For the nuclear reaction given in Eq. (44.2) assume that the initial kinetic energy and momentum of the reacting particles are negligible, Calculate the speed of the $\alpha$ particle immediately after it leaves the reaction region.

Joshua Young

Numerade Educator

Problem 6

Estimate the range of the force mediated by an $\omega^{0}$ meson that has mass $783 \mathrm{MeV} / \mathrm{c}^{2}$.

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Erika Lynn

Numerade Educator

Problem 7

In a collision experiment, a proton at rest is struck by an antiproton. (a) What is the minimum kinetic energy of the antiproton if the available energy is 2.00 TeV? (b) If a colliding beam is used instead of a stationary target, what minimum kinetic energy for each beam is required for an available energy of 2.00 TeV?

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Erika Lynn

Numerade Educator

Problem 8

An electron with a total energy of $30.0 \mathrm{GeV}$ collides with a stationary positron. (a) What is the available energy? (b) If the electron and positron are accelerated in a collider, what total energy corresponds to the same available energy as in part (a)?

Joshua Young

Numerade Educator

Problem 9

Deuterons in a cyclotron travel in a circle with radius $32.0 \mathrm{~cm}$ just before emerging from the dees. The frequency of the applied alternating voltage is $9.00 \mathrm{MHz}$. Find (a) the magnetic field and (b) the kinetic energy and speed of the deuterons upon emergence.

Joshua Young

Numerade Educator

Problem 10

Bubble-chamber photographs taken with a $0.40 \mathrm{~T}$ magnetic field show at one point a positron moving perpendicular to the field in a $15 \mathrm{~cm}$ circle. What is the magnitude of the positron's momentum at that point?

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Erika Lynn

Numerade Educator

Problem 11

(a) A high-energy beam of alpha particles collides with a stationary helium gas target. What must the total energy of a beam particle be if the available energy in the collision is $16.0 \mathrm{GeV} ?$ (b) If the alpha particles instead interact in a colliding-beam experiment, what must the energy of each beam be to produce the same available energy?

Joshua Young

Numerade Educator

Problem 12

The first cyclotron used a $1.3 \mathrm{~T}$ magnetic field and had an $11 \mathrm{~cm}$ radius. (a) Find the maximum kinctic energy of protons accelerated by this cyclotron. Is it accurate to use nonrelativistic expressions in your calculation? (b) When the cyclotron was accelerating protons, at what frequency was it operated?

Jaime Munoz

Numerade Educator

Problem 13

(a) What is the speed of a proton that has total energy $1000 \mathrm{GeV} ?$ (b) What is the angular frequency $\omega$ of a proton with the speed calculated in part (a) in a magnetic field of $4.00 \mathrm{~T}^{2}$ Use both the nonrelativistic Eq. (44.7) and the correct relativistic expression, and compare the results.

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Erika Lynn

Numerade Educator

Problem 14

Calculate the minimum beam energy in a proton-proton collider to initiate the $\mathrm{p}+\mathrm{p} \rightarrow \mathrm{p}+\mathrm{p}+\eta^{0}$ reaction. The rest energy of the $\eta^{0}$ is $547.9 \mathrm{MeV}$ (see Table 44.3 ).

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Erika Lynn

Numerade Educator

Problem 15

You work for a start-up company that is planning to use antiproton annihilation to produce radioactive isotopes for medical applications. One way to produce antiprotons is by the reaction $\mathrm{p}+\mathrm{p} \rightarrow \mathrm{p}+\mathrm{p}+\mathrm{p}+\overline{\mathrm{p}}$ in proton-proton collisions. (a) You first con-sider a colliding-beam experiment in which the two proton beams have cqual kinctic energies. To produce an antiproton via this reaction, what is the required minimum kinetic energy of the protons in each beam? (b) You then consider the collision of a proton beam with a stationary proton target. For this experiment, what is the required minimum kinetic energy of the protons in the beam?

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Erika Lynn

Numerade Educator

Problem 16

In about $8 \%$ of $\Omega^{-}$ decays, the decay products are a $\equiv^{-}$ and a $\pi^{0}$. (a) What is the energy released in this decay? (b) In the decay, what is the change in baryon number, and what is the change in strangeness? Your results should show that this decay is allowed for the weak nuclear interaction hut not for the strone in

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Erika Lynn

Numerade Educator

Problem 17

$ \cdot\mathrm{A} \mathrm{K}^{+}$ meson at rest decays into two $\pi$ mesons. (a) What are the allowed combinations of $\pi^{0}, \pi^{+},$ and $\pi^{-}$ as decay products?
(b) Find the total kinetic energy of the $\pi$ mesons.

Zachary Warner

Numerade Educator

Problem 18

How much energy is released when a $\mu^{-}$ muon at rest decays into an electron and two neutrinos? Neglect the small masses of the neutrinos.

Kai Chen

Princeton University

Problem 19

What is the mass (in kg) of the $Z^{0}$ ? What is the ratio of the mass of the $Z^{0}$ to the mass of the proton?

Kai Chen

Princeton University

Problem 20

The discovery of the $\Omega^{-}$ particle helped confirm Gell-Mann's eightfold way. If an $\Omega^{-}$ decays into a $\Lambda^{0}$ and a $\mathrm{K}^{-}$, what is the total kinetic energy of the decay products?

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Erika Lynn

Numerade Educator

Problem 21

If a $\Sigma^{+}$ at rest decays into a proton and a $\pi^{0},$ what is the total kinetic energy of the decay products?

Kai Chen

Princeton University

Problem 22

Which of the following reactions obey the conservation of baryon number?
(a) $p+p \rightarrow p+e^{+}$
(b) $p+n \rightarrow 2 e^{+}+e^{-}$
(c) $\mathrm{p} \rightarrow \mathrm{n}+\mathrm{e}^{-}+\bar{\nu}_{\mathrm{e}} ;$
(d) $\mathrm{p}+\overline{\mathrm{p}} \rightarrow 2 \gamma$

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Erika Lynn

Numerade Educator

Problem 23

In which of the following decays are the three lepton numbers conserved? In each case, explain your reasoning. (a) $\mu^{-} \rightarrow \mathrm{e}^{-}+v_{e}+\bar{v}_{\mu}$
(b) $\tau^{-} \rightarrow \mathrm{e}^{-}+\bar{\nu}_{e}+\nu_{\tau}$ (c) $\pi^{+} \rightarrow \mathrm{e}^{+}+\gamma_{i}$ (d) $\mathrm{n} \rightarrow \mathrm{p}+\mathrm{e}^{-}+\bar{\nu}_{e}$

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Erika Lynn

Numerade Educator

Problem 24

In which of the following reactions or decays is strangeness conserved? In each case, explain your reasoning.
(a) $\mathrm{K}^{+} \rightarrow \mu^{+}+\nu_{\mu}$
(b) $n+K^{+} \rightarrow p+\pi^{0}$
(c) $\mathrm{K}^{+}+\mathrm{K} \rightarrow \pi^{0}+\pi^{0}$
(d) $p+K^{-} \rightarrow \Lambda^{0}+\pi^{0}$

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Erika Lynn

Numerade Educator

Problem 25

Determine the electric charge, baryon number, strangeness quantum number, and charm quantum number for the following quark combinations: (a) $u d s ;(b) c \bar{u} ;(c) d d d ;$ and $(d) d \bar{c} .$ Explain your reasoning.

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Erika Lynn

Numerade Educator

Problem 26

Determine the electric charge, baryon number, strangeness quantum number, and charm quantum number for the following quark combinations: (a) uus, (b) $c \bar{s},$ (c) $\overline{d d} \bar{u},$ and $($ d) $\bar{c} b$.

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Erika Lynn

Numerade Educator

Problem 27

Given that each particle contains only combinations of $\boldsymbol{u}, \boldsymbol{d}, \boldsymbol{s}, \overline{\boldsymbol{u}}, \overline{\boldsymbol{d}},$ and $\bar{s},$ use the method of Example 44.7 to deduce the quark content of (a) a particle with charge $+e,$ baryon number $0,$ and strangeness $+1 ;$ (b) a particle with charge $+e,$ baryon number -1 and strangeness $+1 ;$ (c) a particle with charge $0,$ baryon number +1 , and strangeness -2 .

Kai Chen

Princeton University

Problem 28

What is the total kinetic energy of the decay products when an upsilon particle at rest decays to $\tau^{+}+\tau^{-} ?$

Tara Appleyard

Numerade Educator

Problem 29

Section 44.5 states that current experiments show that the mass of the Higgs boson is about $125 \mathrm{GeV} / c^{2}$. What is the ratio of the mass of the Higgs boson to the mass of a proton?

Kai Chen

Princeton University

Problem 30

The spectrum of the sodium atom is detected in the light from a distant galaxy. (a) If the $590.0 \mathrm{nm}$ line is redshifted to $658.5 \mathrm{nm},$ at what speed is the galaxy receding from the earth? (b) Use the Hubble law to calculate the distance of the galaxy from the earth.

Kai Chen

Princeton University

Problem 31

A galaxy in the constellation Pisces is 5210 Mly from the earth. (a) Use the Hubble law to calculate the speed at which this galaxy is receding from earth. (b) What redshifted ratio $\lambda_{0} / \lambda_{\mathrm{S}}$ is expected for light from this galaxy?

Ajay Singhal

Numerade Educator

Problem 32

The definition of the redshift $z$ is given in Example $44.8 .$ (a) Show that Eq. (44.13) can be written as $1+z=([1+\beta] /[1-\beta])^{1 / 2},$ where $\beta=v / c .$ (b) The observed redshift for a certain galaxy is $z=0.700$. Find the speed of the galaxy relative to the earth; assume the redshift is described by Eq. (44.14). (c) Use the Hubble law to find the distance of this galaxy from the earth.

Kai Chen

Princeton University

Problem 33

Calculate the reaction energy $Q$ (in MeV) for the reaction $\mathrm{e}^{-}+\mathrm{p} \rightarrow \mathrm{n}+v_{\mathrm{e}} .$ Is this reaction endoergic or exoergic?

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Erika Lynn

Numerade Educator

Problem 34

Calculate the energy (in MeV) released in the triple-alpha process $3^{4} \mathrm{He} \rightarrow{ }^{12} \mathrm{C}$

Ren Jie Tuieng

Numerade Educator

Problem 35

Calculate the reaction energy $Q$ (in $\mathrm{MeV}$ ) for the nucleosynthesis reaction
$${ }_{6}^{12} \mathrm{C}+{ }_{2}^{4} \mathrm{He} \rightarrow{ }_{8}^{16} \mathrm{O}$$
Is this reaction endoergic or exoergic?

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Erika Lynn

Numerade Educator

Problem 36

The strong nuclear force can be crudely modeled as a Hooke's-law spring force, increasing linearly with the quark scparation distance. The energy stored in this "spring" corresponds to the energy content of the gluon field. In this picture, as quarks are further separated, increasing energy is stored between them. At a critical separation distance, the energy converts to matter, and a new quark-antiquark pair is generated, guaranteeing that there can never be a free quark.
(a) A proton has a diameter of about $1.5 \mathrm{fm}$. Fstimate the repulsive Coulomb force between two up quarks separated by $0.5 \mathrm{fm}$. (b) Model the strong force as $F_{\mathrm{s}}=k s,$ where $s$ is the distance between two quarks. If this force balances the electrostatic repulsion between two up quarks when $s=0.5 \mathrm{fm},$ what is the effective spring constant $k,$ in $\mathrm{SI}$ units?
(c) Convert $k$ into units of $\mathrm{MeV} / \mathrm{fm}^{2}$. (d) How much energy is stored in the gluon field when $s=0.5 \mathrm{fm} ?$ The mass of an up quark is thought to be about $2.3 \mathrm{MeV} / \mathrm{c}^{2}$. (e) How much energy is needed to produce an up quark and an antiup quark? (f) How far would two up quarks need to be separated so that the gluon energy $\frac{1}{2} k s^{2}$ matches the rest energy of an up-antiup quark pair?

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Erika Lynn

Numerade Educator

Problem 37

CP BIO Radiation Therapy with $\pi^{-}$ Mesons. Beams of $\pi^{-}$ mesons are used in radiation therapy for certain cancers. The energy comes from the complete decay of the $\pi^{-}$ to stable particles. (a) Write out the complete decay of a $\pi^{-}$ meson to stable particles. What are these particles? (b) How much energy is released from the complete decay of a single $\pi^{-}$ meson to stable particles? (You can ignore the very small masses of the neutrinos.) (c) How many $\pi^{-}$ mesons need to decay to give a dose of 50.0 Gy to $10.0 \mathrm{~g}$ of tissue? (d) What would be the equivalent dose in part (c) in Sv and in rem? Consult Table 43.3 and use the largest appropriate $\mathrm{RBE}$ for the particles involved in this decay.

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Erika Lynn

Numerade Educator

Problem 38

A proton and an antiproton collide head-on with equal kinetic energies. Two $\gamma$ rays with wavelengths of $0.720 \mathrm{fm}$ are produced. Calculate the kinetic energy of the incident proton.

Kai Chen

Princeton University

Problem 39

The 7 TeV proton bunches that circulate in opposite directions around the $27 \mathrm{~km}$ ring of the Large Hadron Collider smash together at beam crossings every 25 ns. (a) How many bunches mect every second? (b) Each bunch has 115 billion protons. Of these, typically 20 collide during each crossing. Estimate the fraction of protons that collide per crossing. (c) Estimate how many collisions take place each second. (d) The bunches are $30 \mathrm{~cm}$ long and are squeezed to a diameter of $20 \mu \mathrm{m}$. Estimate the density of protons in a bunch, in units of protons/mm $^{3}$. (c) Estimate the density of hadrons in ordinary matter. (Hint: Divide your mass, which is mostly due to hadrons, by the mass of a proton to get the number of hadrons in your body. Then divide by the estimated volume.)

Eduard Sanchez

Numerade Educator

Problem 40

Calculate the threshold kinetic energy for the reaction $\pi^{-}+p \rightarrow \Sigma^{0}+K^{0}$ if a $\pi^{-}$ beam is incident on a stationary proton target. The $\mathrm{K}^{0}$ has a mass of $497.7 \mathrm{MeV} / \mathrm{c}^{2}$.

Kai Chen

Princeton University

Problem 41

Each of the following reactions is missing a single particle. Calculate the baryon number, charge, strangeness, and the three lepton numbers (where appropriate) of the missing particle, and from this (b) $\mathrm{K}^{-}+\mathrm{n} \rightarrow \Lambda^{0}+?$ identify the particle. (a) $p+p \rightarrow p+\Lambda^{0}+?$ (c) $p+\bar{p} \rightarrow n+?$ (d) $\bar{v}_{\mu}+\mathbf{p} \rightarrow \mathrm{n}+?$

Eduard Sanchez

Numerade Educator

Problem 42

An $\eta^{0}$ meson at rest decays into three $\pi$ mesons. (a) What are the allowed combinations of $\pi^{0}, \pi^{+}$, and $\pi^{-}$ as decay products? (b) Find the total kinetic energy of the $\pi$ mesons.

Kai Chen

Princeton University

Problem 43

The $\phi$ meson has mass $1019.4 \mathrm{MeV} / \mathrm{c}^{2}$ and a measured energy width of $4.4 \mathrm{MeV} / \mathrm{c}^{2}$. Using the uncertainty principle, estimate the lifetime of the $\phi$ meson.

Kai Chen

Princeton University

Problem 44

Estimate the energy width (energy uncertainty) of the $\psi$ if its mean lifetime is $7.6 \times 10^{-21}$ s. What fraction is this of its rest energy?

Kai Chen

Princeton University

Problem 45

One proposed proton decay is $\mathrm{p}^{+} \rightarrow \mathrm{e}^{+}+\pi^{0}$, which violates both baryon and lepton number conservation, so the proton lifetime is expected to be very long. Suppose the proton half-life were $1.0 \times 10^{18} \mathrm{y} .$ (a) Calculate the energy deposited per kilogram of body tissue (in rad) due to the decay of the protons in your body in one year. Model your body as consisting entirely of water. Only the two protons in the hydrogen atoms in each $\mathrm{H}_{2} \mathrm{O}$ molecule would decay in the manner shown; do you see why? Assume that the $\pi^{0}$ decays to two $\gamma$ rays, that the positron annihilates with an electron, and that all the encrgy produced in the primary decay and these secondary decays remains in your body. (b) Calculate the equivalent dose (in rem) assuming an RBE of 1.0 for all the radiation products, and compare with the 0.1 rem due to the natural background and the 5.0 rem guideline for industrial workers. Based on your calculation, can the proton lifetime be as short as $1.0 \times 10^{18} \mathrm{y} ?$

Keshav Singh

Numerade Educator

Problem 46

$\mathrm{A} \phi$ meson (see Problem 44.43 ) at rest decays via $\phi \rightarrow \mathrm{K}^{+}+\mathrm{K}$. It has strangeness $0 .$ (a) Find the kinetic energy of the $\mathrm{K}^{+}$ meson. (Assume that the two decay products share kinetic energy equally, since their masses are equal.) (b) Suggest a reason the decay $\phi \rightarrow \mathrm{K}^{+}+\mathrm{K}^{-}+\pi^{0}$ has not been observed.
(c) Suggest reasons the decays $\phi \rightarrow \mathrm{K}^{+}+\pi^{-}$ and $\phi \rightarrow \mathrm{K}^{+}+\mu^{-}$ have not been observed.

Kai Chen

Princeton University

Problem 47

About 10,000 cosmic-ray protons, each with hundreds of MeV of energy, impinge on each square meter of our upper atmosphere each second. They collide with atmospheric nitrogen and oxygen to produce secondary showers of newly created particles, including many muons. Muons have a mass of $105.7 \mathrm{MeV} / c^{2}$ and an average lifetime of $2.197 \mu \mathrm{s}$. Consider a secondary cosmic muon produced at an altitude of $15.00 \mathrm{~km}$ aimed directly downward with an energy of $6.000 \mathrm{GeV}$. With such high energy, the muon can travel a great distance into the earth without slowing down. (a) Determine the speed of this muon. (b) How far would this muon travel in one lifetime if there were no relativistic effects? (c) In the frame of the muon, what distance separates its creation position from the earth's surface? (d) If one lifetime passes in the frame of the muon, how much time passes in the frame of the earth? (e) How far does a muon travel in this time as seen from the earth? (f) Does the muon survive its trip to the surface? How far will it penetrate the earth in its lifetime?

Keshav Singh

Numerade Educator

Problem 48

A particle at rest decays to a $\Lambda^{0}$ and a $\pi^{-}$. (a) Find the total kinetic energy of the decay products. (b) What fraction of the energy is carried off by each particle? (Use relativistic expressions for momentum and energy.)

Keshav Singh

Numerade Educator

Problem 49

A $\Sigma^{-}$ particle moving in the $+x$ -direction with kinetic energy $180 \mathrm{MeV}$ decays into a $\pi^{-}$ and a neutron. The $\pi^{-}$ moves in the $+y$ -direction. What is the kinetic energy of the neutron, and what is the direction of its velocity? Use relativistic expressions for energy and momentum.

Kai Chen

Princeton University

Problem 50

The $\mathrm{K}^{0}$ meson has rest energy $497.7 \mathrm{MeV}$. A $\mathrm{K}^{0}$ meson moving in the $+x$ -direction with kinetic energy $225 \mathrm{MeV}$ decays into a $\pi^{+}$ and a $\pi^{-}$, which move off at equal angles above and below the $+x$ -axis. Calculate the kinetic energy of the $\pi^{+}$ and the angle it makes with the $+x$ -axis. Use relativistic expressions for energy and momentum.

Keshav Singh

Numerade Educator

Problem 51

While tuning up a medical cyclotron for use in isotope production, you obtain the data given in the table. $$ \begin{array}{l|llll} B(\mathrm{~T}) & 0.10 & 0.20 & 0.30 & 0.40 \\
\hline K_{\max }(\mathrm{MeV}) & 0.068 & 0.270 & 0.608 & 1.080 \end{array} $$
$B$ is the uniform magnetic field in the cyclotron, and $K_{\max }$ is the maximum kinetic energy of the particle being accelerated, which is a proton. The radius $R$ of the proton path at maximum kinetic energy has the same value for each magnetic-field value. (a) Compare the kinetic energy values in the table to the rest energy $m c^{2}$ of a proton. Is it necessary to use relativistic expressions in your analysis? Explain. (b) Graph your data as $K_{\max }$ versus $B^{2}$. Use the slope of the best-fit straight line to your data to find $R$. (c) What is the maximum kinetic energy for a $0.25 \mathrm{~T}$ magnetic field? (d) What is the angular frequency $\omega$ of the proton when $B=0.40 \mathrm{~T} ?$

Keshav Singh

Numerade Educator

Problem 52

The decay products from the decay of short-lived unstable particles can provide evidence that these particles have been produced in a collision experiment. As an initial step in designing an experiment to detect short-lived hadrons, you make a literature study of their decays. Table 44.3 gives experimental data for the mass and typical decay modes of the particles $\Sigma^{-}, \Xi^{0}, \Delta^{++},$ and $\Omega^{-}$
(a) Which of these four particles has the largest mass? The smallest? (b) By the decay modes shown in the table, for which of these particles do the decay products have the greatest total kinetic energy? The least?

Keshav Singh

Numerade Educator

Problem 53

DATA You have entered a graduate program in particle physics and are learning about the use of symmetry. You begin by repeating the analysis that led to the prediction of the $\Omega^{-}$ particle. Nine of the spin- $\frac{3}{2}$ baryons are four $\Delta$ particles, each with mass $1232 \mathrm{MeV} / c^{2}$. strangeness $0,$ and charges $+2 e,+e, 0,$ and $-e ;$ three $\Sigma^{\circ}$ particles, each with mass $1385 \mathrm{MeV} / \mathrm{c}^{2}$, strangeness -1 , and charges $+e .0,$ and $-e$ and two $\equiv=$ particles, each with mass $1530 \mathrm{MeV} / \mathrm{c}^{2},$ strangeness $-2,$ and charges 0 and $-e .$ (a) Place these particles on a plot of $S$ versus Q. Deduce the $Q$ and $S$ values of the tenth spin- $\frac{1}{2}$ baryon, the $\Omega^{-}$ particle, and place it on your diagram. Also label the particles with their masses. The mass of the $\Omega^{-}$ is $1672 \mathrm{MeV} / \mathrm{c}^{2} ;$ is this value consistent with your diagram? (b) Deduce the three-quark combinations (of $u, d$, and I s) that make up each of these ten particles. Redraw the plot of $S$ versus $Q$ from part (a) with cach particle labeled by its quark content. What regularities do you see?

Keshav Singh

Numerade Educator

Problem 54

A positive kaon $K^{+}$ which is a bound state of an up quark and an antidown quark, can decay into an antimuon and a muon neutrino via a weak interaction process mediated by a $W^{+}$ boson. We can determine how much of the kaon's energy each of the daughter particles obtains using relativistic kinematics. (a) In the rest frame of the kaon, the speed of the antimuon is $v$ and its Lorentz factor is $\gamma=\left(1-v^{2} / c^{2}\right)^{-1 / 2}$. Using the notation $E_{v}$ for the energy of the neutrino, $M_{K}$ for the mass of the kaon, and $M_{\mu}$ for the mass of the antimuon, write the relativistic expression for conservation of energy. Treat the neutrino as massless. (b) Write the relativistic equation for conservation of momentum. (c) Solve these two equations to determine $v$ as a function of the mass ratio $\sigma=M_{K} / M_{\mu}$. (d) Using the masses $M_{K}$ and $M_{\mu}$ found in Tables 44.3 and 44.2 , respectively, determine the value of $\sigma$. (e) What is the energy of $E_{\mu}=\gamma M_{\mu} c^{2}$ of the antimuon? (f) What is the energy of the neutrino? (g) Do these energies add up to the rest energy of the kaon?

Keshav Singh

Numerade Educator

Problem 55

Consider the following hypothetical universe: The Nibiruvians and the
Xibalbans are minuscule "flat" societies located at an angular separation of $\theta=60^{\circ}$ on the surface of a two-dimensional spherical universe with radius $R$, as shown in Fig. $P 44.55$. (a) What is the distance $D$ between the two societies in terms of $R$ and $\theta ?$ (b) If $R$ is increas- ing in time, what is the speed $V=d D / d t$ at which the civilizations are separating. as a function of $R, d R / d t,$ and $D ?$ (c) The separation velocity and the distance $D$ are related by $\underline{V}=B D$. Determine the "Bubble parameter" $B$ in terms of $R(t) .$ Note that $B$ is not constant. (Similarly, the Hubble "constant" $H_{0}$ is presumed to be changing with time.) (d) The radius of this universe was $R_{0}=500.0 \mathrm{~m}$ upon its creation at time $t=0,$ and it has been increasing at a constant rate of $1.00 \mu \mathrm{m} / \mathrm{s}$. What was the Bubble parameter exactly four years after creation? (e) What is the distance between Nibiru and Xibalba four years after creation? (f) At what speed are they separating at that time? (g) At that time, the Nibiruvians transmit isotropic ripple waves with their "light speed" of $c=6.35 \mu \mathrm{m} / \mathrm{s}$ and wavelength $1.00 \mathrm{nm}$. How long does it take these waves to reach Xibalba? (Hint: In time $d t$ the waves travel an angular distance $d \theta=c d t / R(t) .$ Integrate to obtain an expression for the total angular distance traveled as a function of time. Solve for the time needed to travel a given angular distance.) (h) At what wavelength will the Xibalbans observe these waves? (Hint:
Use Eq. $(44.16) .)$

Keshav Singh

Numerade Educator

Problem 56

Consider a collision in which a stationary particle with mass $M$ is bombarded by a particle with mass $m,$ speed $t_{0}$. and total energy (including rest cnergy) $E_{m}$. (a) Use the Lorentz transformation to write the velocities $v_{m}$ and $v_{M}$ of particles $m$ and $M$ in terms of the speed $v_{\mathrm{cm}}$ of the center of momentum. (b) Use the fact that the total momentum in the center-of-momentum frame is zero to obtain an expression for $v_{\mathrm{cm}}$ in terms of $m, M,$ and $v_{0}$. (c) Combine the results of parts (a) and (b) to obtain Eq. (44.9) for the total energy in the center-of-momentum frame.

Keshav Singh

Numerade Educator

Problem 57

What is the energy of each photon produced by positronelectron annihilation? (a) $\frac{1}{2} m_{e} v^{2},$ where $v$ is the speed of the emitted positron; (b) $m_{e} v^{2} ;(c) \frac{1}{2} m_{e} c^{2} ;$ (d) $m_{e} c^{2}$.

Eduard Sanchez

Numerade Educator

Problem 58

Suppose that positron-electron annihilations occur on the line $3 \mathrm{~cm}$ from the center of the line connecting two detectors. Will the resultant photons be counted as having arrived at these detectors simultancously? (a) No, because the time difference between their arrivals is $100 \mathrm{~ms} ;$ (b) no, because the time difference is $200 \mathrm{~ms} ;$ (c) yes, because the time difference is $0.1 \mathrm{~ns} ;$ (d) yes, because the time difference is $0.2 \mathrm{~ns}$.

Eduard Sanchez

Numerade Educator

Problem 59

If the annihilation photons come from a part of the body that is separated from the detector by $20 \mathrm{~cm}$ of tissue, what percentage of the photons that originally traveled toward the detector remains after they have passed through the tissue?
(a) $1.4 \% ;$ (b) $8.6 \% ;$ (c) $14 \% ;$ (d) $86 \%$.

Bettina Hanlon

Numerade Educator