Съдържание

IOAA 2014, Теория, предварителен вариант — Задача 3. Cosmic radiation

Оригинал на условието (PDF)Съобщи за грешка

Всички материали от IOAA 2014

8th International Olympiad on Astronomy and Astrophysics, Suceava - Gura Humorului - August 2014 — THEORETICAL TEST, 15 short problems

Indications

The problems were elaborated concerning two aspects: a) To cover merely all the subjects from the syllabus;
b) The average time for solving the items is about 15 minutes per a short problem;
In your folder you will find out the following: a) Answer sheets
b) Draft sheets
c) The envelope with the subjects in English and the translated version of them in your mother tongue;
The solutions of the problems will be written down only on the answer sheets you receive on your desk. PLEASE WRITE ONLY ON THE PRINTED SIDE OF THE PAPER SHEET. DON’T USE THE REVERSE SIDE. The evaluator will not take into account what is written on the reverse of the answer sheet.
The draft sheets is for your own use to try calculation, write some numbers etc. BEWARE: These papers are not taken into account in evaluation, at the end of the test they will be collected separately . Everything you consider as part of the solutions have to be written on the answer sheets.
Each problem have to be started on a new distinct answer sheet.
On each answer sheet please fill in the designated boxes as follows: a) In PROBLEM NO. box write down only the number of the problem: i.e. 1 – 15 for short problems, 16 – 19 for long problems. Each sheet containing the solutions of a certain problem, should have in the box the number of the problem;
b) In Student ID – fill in your ID you will find on your envelope, consisted of 3 letters and 2 digits.
c) In page no. box you will fill in the number of page, starting from 1. We advise you to fill this boxes after you finish the test
We don’t understand your language, but the mathematic language is universal, so use as more relationships as you think that your solution will be better understand by the evaluator. If you want to explain in words we kindly ask you to use short English propositions.
Use the pen you find out on the desk. It is advisable to use a pencil for the sketches.
At the end of the test: a) Don’t forget to put in order your papers;
b) Put the answer sheets in the folder 1. Please verify that all the pages contain your ID, correct numbering of the problems and all pages are in the right order and numbered. This is an advantage of ease of understanding your solutions.
c) Verify with the assistant the correct number of answer sheets used fill in this number on the cover of the folder and sign.
d) Put the draft papers in the designated folder, Put the test papers back in the envelope.
e) Go to swim

GOOD LUCK !

Условие

During studies concerning cosmic radiation, a neutral unstable particle – the $\pi^0$ meson was identified. The rest-mass of meson $\pi^0$ is much larger than the rest-mass of the electron. The studies reveal that during its flight, the meson $\pi^0$ disintegrates into 2 photons.

Find an expression for the initial velocity of the meson $\pi^0$ , if after its disintegration, one of the photons has the maximum possible energy $E_{\max}$ and, consequently, the other photon has the minimum possible energy $E_{\min}$ . You may use as known c - the speed of light.

Отговори

Решение

Problem 3. Marking scheme Cosmic radiation

Correct use of general laws of conservation (A) — 5 points
Correct applying of the laws of conservation for the conditions stated in the problem (B) — 4 points
Correct conduct of calculations and final solution (C) — 1 point

Detailed solution

(A)

In the disintegration process the laws of energy conservation and the law of the conservation of momentum are both obeyed.

In the general case the law of conservation of the momentum is represented in the down below figure.

Left: the meson π⁰ (labelled “mesone π⁰”) moving right with momentum p. Right: after the decay, photons f₁ and f₂ with momenta p₁ and p₂ at angles θ₁ and θ₂ to the dashed direction of p, completing a dashed parallelogram whose diagonal is p.

the total initial energy of the $\pi^0$ meson is

E^2 = p^2c^2 + m_0^2c^4

And its kinetic energy is

E_c = E - m_0c^2

The expressions of the 2 conservation laws written after the disintegration are:

\vec{p} = \vec{p}_1 + \vec{p}_2;

E + m_0c^2 = E_1 + E_2,

The energy of the photon 1 can be calculated using the notations in the figure

E = E_c + m_0c^2;

p_1\sin\theta_1 = p_2\sin\theta_2;

\frac{E_1}{c}\sin\theta_1 = \frac{E_2}{c}\sin\theta_2;

E^2 = p^2c^2 + m_0^2c^4;

p^2c^2 = E^2 - m_0^2c^4;

E = E_1 + E_2;

E_2 = E - E_1 = \left(E_c + m_0c^2\right) - E_1;

E_1 = \frac{{m_0}^2c^4}{2}\frac{1}{E_c + m_0c^2 - \cos\theta_1\sqrt{E_c\left(E_c + m_0c^2\right)}}.

Similar the second photon energy is:

E_2 = \frac{{m_0}^2c^4}{2}\frac{1}{E_c + m_0c^2 - \cos\theta_2\sqrt{E_c\left(E_c + m_0c^2\right)}}

(B)

If one of the photon has the maximum possible energy $E_{\max}$ and consequently the other photon has the minimum possible energy $E_{\min}$ the law of momentum conservation is sketched:

Collinear case: the meson π⁰ of mass m moving right with velocity v; photon f₂ with momentum p_min pointing left and photon f₁ with momentum p_max pointing right along the same line.

Thus the relations become very simple:

mv = \frac{m_0\mathbf{v}}{\sqrt{1 - \frac{\mathbf{v}^2}{c^2}}}

E_{\max} = \frac{m_0c^2}{2\sqrt{1 - \frac{\mathbf{v}^2}{c^2}}}\left(1 + \frac{\mathbf{v}}{c}\right);

E_{\min} = \frac{m_0c^2}{2\sqrt{1 - \frac{\mathbf{v}^2}{c^2}}}\left(1 - \frac{\mathbf{v}}{c}\right);

v = c\frac{E_{\max} - E_{\min}}{E_{\max} + E_{\min}}. \qquad \textbf{(C)}

Оригинал в Архива: th_short_ver1.pdf · официални решения: th_short_ver1_sol.pdf