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Условие Решение

USAPhO 2000, Полуфинал — A3

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Всички материали от USAPhO 2000

2000 Semi-Final Exam · 20 т.

Указания

DO NOT OPEN THIS TEST UNTIL YOU ARE TOLD TO BEGIN

Work Part A first. You have 90 minutes to complete all four problems.
After you have completed Part A, you may take a break.
Then work Part B. You have 90 minutes to complete both problems.
Show all your work. Partial credit will be given.
Start each question on a new sheet of paper. Be sure to put your name in the upper right-hand corner of each page, along with the question number and the page number/total pages for this problem. For example,

Doe, Jamie

A1 – 1/3

A hand-held calculator may be used. Its memory must be cleared of data and programs. It may not be used in graphing mode. Calculators may not be shared. You may not use any tables, books, or collections of formulas. You may use a ruler or straight edge.
Questions with the same point value are not necessarily of the same difficulty.
Good luck!

Possibly Useful Information

Quantity	Value
Gravitational field at the Earth's surface	$g=9.8$ N/kg
Newton's gravitational constant	$G=6.67\times10^{-11}$ N·m $^2$ /kg $^2$
Coulomb's constant	$k=1/4\pi\epsilon_0=9.0\times10^9$ N·m $^2$ /C $^2$
Biot-Savart constant	$k_m=\mu_0/4\pi=10^{-7}$ T·m/A
Speed of light in a vacuum	$c=3.0\times10^8$ m/s
Boltzmann's constant	$k_B=1.38\times10^{-23}$ J/K
Avogadro's number	$N_A=6.02\times10^{23}$ (mol) $^{-1}$
Ideal gas constant	$R=N_Ak_B=8.31$ J/mol·K
Elementary charge	$e=1.6\times10^{-19}$ C
1 electron volt	$1\text{ eV}=1.6\times10^{-19}$ J
Planck's constant	$h=6.63\times10^{-34}\text{ J}\cdot\text{s}=4.14\times10^{-15}\text{ eV}\cdot\text{s}$
Electron mass	$m=9.1\times10^{-31}\text{ kg}=0.51\text{ MeV}/c^2$
Binomial expansion	$(1+x)^n\simeq1+nx\quad\text{for }\lvert x\rvert\ll1$

Условие

A relativistic particle decays into two photons. One of the photons is traveling to the right along the x-axis with frequency $\nu_1$ . The second photon is traveling to the left with frequency $\nu_2$ , where $\nu_1>\nu_2$ . Express your answers to the following in terms of given quantities and known constants.

a. Find the velocity $V$ of the original particle. [5 т.]

b. Find the rest mass of the original particle. [5 т.]

c. What are the frequencies of the photons in the rest frame of the original particle? What are their directions of travel in this rest frame? [5 т.]

d. You are given the following formula:

p'_x=F_1p_x+F_2\frac{E_\gamma}{c}

where $p'_x$ is the observed x-component of the momentum of either photon, whereas $E_\gamma$ and $p_x$ are the energy and x-component of the momentum of the photon in the rest frame of the particle. Using your results determine the functions $F_1$ and $F_2$ in terms of $V/c$ , where $V$ is the speed of the original particle. [5 т.]

Решение

a. (5 т.)

The relativistic energy and momentum of a particle are given by

E=\frac{mc^2}{\sqrt{1-(V/c)^2}},\qquad p=\frac{mV}{\sqrt{1-(V/c)^2}}

where $m$ is the particle mass and $V$ is the particle speed. The energy and momentum of a photon of frequency $\nu$ are given by

E_\gamma=h\nu,\qquad p_\gamma=\frac{h\nu}{c}.

Applying energy conservation, we have

\frac{mc^2}{\sqrt{1-(V/c)^2}}=h\nu_1+h\nu_2\qquad (A3.1)

while momentum conservation gives

\frac{mV}{\sqrt{1-(V/c)^2}}=\frac{h\nu_1}{c}-\frac{h\nu_2}{c}.\qquad (A3.2)

Dividing (A3.2) by (A3.1), we have

\frac{mV}{mc^2}=\frac{h(\nu_1-\nu_2)}{ch(\nu_1+\nu_2)}.

Canceling common factors and solving for $V$

V=c\frac{\nu_1-\nu_2}{\nu_1+\nu_2}\quad\text{to the right}.\qquad (A3.3)

b. (5 т.)

Solving (A3.1) for $m$

m=\frac h{c^2}(\nu_1+\nu_2)\sqrt{1-(V/c)^2}.

and substituting in (A3.3)

m=\frac h{c^2}(\nu_1+\nu_2)\sqrt{1-\left(\frac{\nu_1-\nu_2}{\nu_1+\nu_2}\right)^2}=\frac h{c^2}\sqrt{(\nu_1+\nu_2)^2-(\nu_1-\nu_2)^2}.

m=\frac{2h}{c^2}\sqrt{\nu_1\nu_2}.\qquad (A3.4)

c. (5 т.)

In the rest frame of the particle the two photons must have equal and opposite momenta. 1 travels to the right with frequency $\nu$ and 2 travels to the left with frequency $\nu$ . In its rest frame the particle’s energy is $mc^2$ . Applying energy conservation we have

mc^2=h\nu+h\nu=2h\nu.

Combining this with (A3.4)

mc^2=2h\sqrt{\nu_1\nu_2}=2h\nu.

\nu=\sqrt{\nu_1\nu_2}=\frac{mc^2}{2h}

d. (5 т.)

Writing the given equation for photon 1

\frac{h\nu_1}{c}=F_1\frac{h\nu}{c}+F_2\frac{h\nu}{c}=(F_1+F_2)\frac{h\nu}{c}

(F_1+F_2)=\frac{2h\nu_1}{mc^2}.\qquad (A3.5)

While for photon 2

-\frac{h\nu_2}{c}=-F_1\frac{h\sqrt{\nu_1\nu_2}}{c}+F_2\frac{h\sqrt{\nu_1\nu_2}}{c}

(F_1-F_2)=\frac{2h\nu_2}{mc^2}\qquad (A3.6)

Adding (A3.5) and (A3.6)

2F_1=\frac{2h}{mc^2}(\nu_1+\nu_2)

and comparing to (A3.1)

F_1=\frac1{\sqrt{1-(V/c)^2}}.

Subtracting (A3.6) from (A3.5) we have

2F_2=\frac{2h}{mc^2}(\nu_1-\nu_2)

and comparing to (A3.2)

F_2=\frac{(V/c)}{\sqrt{1-(V/c)^2}}

Оригинал в Архива: Semifinal 2000.pdf · официални решения: Semifinal 2000 Solutions.pdf