Съдържание

Условие Отговори Решение

IOAA 2014, Теория, окончателен вариант — Задача 4. Sandra Bullock And George Cloony

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Всички материали от IOAA 2014

8th International Olympiad on Astronomy and Astrophysics — Theoretical Test, 12 short problems (final edition)

Указания

In your folder you will find the following items:

a. Answer sheets

b. Rough work sheets

c. The envelope with the problems The solutions of the problems should be written only on the answer sheets you receive. 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 side of the answer sheet.
The rough worksheets are for your own use for doing calculations, write some numbers etc. BEWARE: These sheets are not taken into account for the evaluation. At the end of the test they will be collected separately. Everything you consider as part of the solutions should be written on the answer sheets.
Each problem should be started on a separate answer sheet.
On each answer sheet please fill in the designated boxes as follows:

a. In the „PROBLEM NO.” box write down only the number of the problem: i.e. 1 – 12 for each short problems, 13 – 15 for each 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 that you will find on your envelope, consisting of 3 letters and 2 digits.

c. In the „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 language of Mathematics is universal, so, please, use as many mathematical expressions as you think that may help the evaluator to better understand your solutions. If you want to explain something in words we kindly ask you to use short phrases(if possible in English).
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:

d. Don’t forget to put your papers in order.

e. Put the answer sheets in 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 will help us to understanding your solutions.

f. Verify with the assistant the correct number of answer sheets used and fill in this number on the cover of the folder and sign it.

g. Put the draft papers in the designated folder. Put the test papers back in the envelope.

h. Go to swim

GOOD LUCK !

Marking scheme

ALL THE CORRECT SOLUTIONS WHICH ARE DIFFERENT FROM THE AUTHOR’S SOLUTION WILL BE EVALUATED AND MARKED ACORDINGLY TO THE MARKING SCHEME.

Условие

An astronaut, with mass $M=100\ \mathrm{kg}$ , gets out of the space ship for a repairing mission. He has to repair a satellite at rest relative to the space ship, at about $d=90\ \mathrm{m}$ away from it. After he finishes his job, he realizes that the systems designed to assure his come-back to shuttle are broken. He also observes that he has air only for 3 minutes. He also notices that he possessed a sealed cylindrical can (base section $S=30\ \mathrm{cm^2}$ ) firmly attached to his/her glove, with $m=200\ \mathrm{g}$ of ice inside. The can is not completely filled with ice.

Determine if the astronaut is able to return safely to the shuttle, before his air reserve is empty, if he manages to open the can in correct direction. Briefly explain your calculations. Note that he cannot throw away anything of its equipment, or touch the satellite.

You may use the following data: $T=272\ \mathrm{K}$ / the temperature of the ice in the can, $p_s=550\ \mathrm{Pa}$ - the pressure of the saturated water vapors at the temperature $T=272\ \mathrm{K}$ ; $R=8300\ \mathrm{J/(kmol\cdot K)}$ - the universal gas constant; $\mu=18\ \mathrm{kg/kmol}$ - the molar mass of the water.

Отговори

Решение

Problem 4. Marking scheme The Astronaut saved by … ice from a can!

Criterion	Points
A. For the use with an adequate justify of one of the relationships (4)	3
B. Reasoning The student describe correctly the processes before and after the can is opened.	4
C. Calculations according to the reasoning, and/or as support for reasoning	2
D. Correct result	1

Detailed solution

Theoretical considerations:

\Omega=\frac{\Delta N}{\Delta t}=\frac{N/6}{\Delta t}=\frac{N/6}{a/\bar v}=\frac16\frac{N\bar v}{a},

\bar v=\sqrt{\frac{3RT}{\mu}},\quad N=n\cdot a^3,

\Omega=\frac16\frac{n\cdot a^3\bar v}{a}=\frac16n\cdot a^2\cdot\bar v;

a^2=S;\quad\Omega=\frac16n\cdot S\cdot\bar v;

\Phi=\frac{\Delta m}{\Delta t},\quad\Phi=\frac{\Delta m}{\Delta t}=\frac{m_0\Delta N}{\Delta t},

\Phi=m_0\Omega=\frac16m_0nS\bar v;

m_0n=m_0\frac{N}{a^3}=\frac{m_0N}{a^3}=\frac mV=\rho,\quad\rho=\frac{\mu p}{RT},

\Phi=\frac16\rho S\bar v=\frac16\frac{\mu p}{RT}S\sqrt{\frac{3RT}{\mu}}=\frac16pS\sqrt{\frac{3\mu}{RT}}.

B. Reasoning

Because the cylindrical can is not full of ice, in the empty part of it there are saturated vapors, i.e the mass flux of the molecule which sublimate is equal with mass flux of gas which transform into ice. Thus the pressure in the can is the saturated vapor pressure $p_s$ and it has the corresponding maximum density $\rho_s$ See figure 6.2

Fig. 6.2. Can with ice and saturated vapour before and after opening, with opposing mass fluxes and pressure labels.

\Phi_1=\Phi_2.

\Phi_1 = \Phi_{sublimation} = \frac{1}{6}\cdot\rho_s\cdot S\cdot\bar{v} = \frac{1}{6}\cdot p_s\cdot S\cdot\sqrt{\frac{3\mu}{RT}};

\Phi_2 = \Phi_{solidification} = \frac{1}{6}\cdot\rho_s\cdot S\cdot\bar{v} = \frac{1}{6}\cdot p_s\cdot S\cdot\sqrt{\frac{3\mu}{RT}};

After the can was opened, there will be no molecules which desublimate thus the mass flux of the molecules which gather the ice become null. So the pressure becomes $\left(p_s/2\right)$ . 2p

C. Calculations according to the reasoning, and/or as support for reasoning

F = \frac{p_s}{2}\cdot S,

a = \frac{F}{M} = \frac{p_s\cdot S}{2M} = \frac{550\ \mathrm{Nm^{-2}}\cdot 30\cdot 10^{-4}\ \mathrm{m^2}}{2\cdot 10^2\ \mathrm{kg}} = 0{,}00825\ \mathrm{ms^{-2}}.

The total time of the acceleration movement will be the total time of ice sublimation:

\tau = \frac{m}{\Phi_1} = \frac{m}{\frac{1}{6}\cdot p_s\cdot S\cdot\sqrt{\frac{3\mu}{RT}}} = \frac{6m}{p_s\cdot S}\cdot\sqrt{\frac{RT}{3\mu}} \approx 150\ \mathrm{s}.

D. Correct result (1 т.)

The travel distance in this time will be :

L = \frac{a\tau^2}{2} = \frac{0{,}00825\ \mathrm{ms^{-2}}\cdot 225\cdot 10^2\ \mathrm{s^2}}{2} \approx 93\ \mathrm{m},

The astronaut could arrive safely in at the shuttle if he didn’t lose to much time by solving the problem.

Оригинал в Архива: Short Theoretical.pdf · официални решения: short_final_sol.pdf · допълнителни данни: ioaa1.jpg · допълнителни данни: ioaa2.jpg · допълнителни данни: ioaa3.jpg · допълнителни данни: ioaa4.jpg · допълнителни данни: ioaa5.jpg · допълнителни данни: ioaa6.jpg · допълнителни данни: ioaa7.jpg