Съдържание

IAO 2008, Теория, Група β — Задача 4. Jump of bear.

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Всички материали от IAO 2008

XIII International Astronomy Olympiad · Theoretical round. Problems to solve · 13 октомври 2008 г.

Бележка

Maybe not all problems have correct questions. Some questions (maybe the main question of the problem, maybe one of the subquestions) may have no real sense. In this case you have to write in your answer (in English or Russian): «situation is impossible – ситуация невозможна». Of course, this answer has to be explained numerically or logically.
Data from the table of planetary data may be used for solving every problem.

Note for jury and team leaders.

The proposed sketches are not full; the team leaders have to give more detailed explanations for students. But the correct solutions in the students’ papers (enough for 8 pts) may be shorter.

Элементы орбит. Физические характеристики некоторых планет, Луны и Солнца

Parameters of orbits.
Physical characteristics of some planets, Moon and Sun

Небесное тело, планета / Body, planet	Среднее расстояние от центрального тела / Average distance to central body, астр. ед. / astr. units	Среднее расстояние от центрального тела / Average distance to central body, млн. км / mln. km	Сидерический (или аналогичный) период обращения / Sidereal (or analogous) period, тропич. годах / troph. years	Сидерический (или аналогичный) период обращения / Sidereal (or analogous) period, средних сутках / days	Эксцентриситет / Eccentricity, e	Экваториальн. диаметр / Equat. diameter, km	Масса / Mass, $10^{24}$ kg	Средняя плотность / Av. density, g/cm³	Ускор. своб. пад. у пов. / Grav. acceler. at surf., m/s²	Макс. блеск, вид. с Земли / Max. magn. from Earth **)	Альбедо / Albedo
Солнце / Sun	1,6·10⁹	2,5·10¹¹	2,2·10⁸	8·10¹⁰		1392000	1989000	1,409		−26,8ᵐ
Меркурий / Mercury	0,387	57,9	0,241	87,97	0,206	4 879	0,3302	5,43	3,70		0,06
Венера / Venus	0,723	108,2	0,615	224,70	0,007	12 104	4,8690	5,24	8,87		0,78
Земля / Earth	1,000	149,6	1,000	365,26	0,017	12 756	5,9742	5,515	9,81		0,36
Луна / Moon	0,00257	0,38440	0,0748	27,3217	0,055	3 475	0,0735	3,34	1,62	−12,7ᵐ	0,07
Марс / Mars	1,524	227,9	1,880	686,98	0,093	6 794	0,6419	3,94	3,71		0,15
Юпитер / Jupiter	5,204	778,6	11,862	4 332,59	0,048	142 984	1899,8	1,33	24,86		0,66
Сатурн / Saturn	9,584	1433,7	29,458	10 759,20	0,054	120 536	568,50	0,70	10,41		0,68

**) Для Луны – в среднем противостоянии.
**) For Moon – in mean opposition.

Координаты на эпоху 2000
Epoch 2000 coordinates

	α	δ
Полярная звезда / Polar star	2ʰ 31ᵐ 48ˢ.7	89° 15′ 51″

Условие

The beginning of XXI century. One may find in guidebooks about Spitsbergen a phrase that “a polar bear jumps 8 meters without warning”. The middle of XXVI century. In order for settling remote areas of Solar system by fauna, biologists plan to deliver polar bears from Spitsbergen onto ice asteroids of the Kuiper belt. However, physicists warn, that some jumping bears can become independent objects of the Kuiper belt. Estimate, on asteroids of what sizes (diameters) it is possible to place polar bears comfortably from the physicists’ point of view. You should give an answer in form of formula-inequality.

Polar bear leaping between ice floes. — Jump of polar bear

Отговори

Решение

4αβ. Jump of bear. So, the bear makes a start and flies... On Spitsbergen (that is, on the Earth) the length of jump is equal to

L=V_{hor}\times t,\qquad t=2V_{vert}/g,

L=2V_{hor}V_{vert}/g.

V_{hor}=V_0\cos\alpha,\qquad V_{vert}=V_0\sin\alpha.

$V_0$ here is the initial speed of jump of the bear from the surface, $g$ – acceleration of gravity at the Earth. If we consider that this speed is not depend on the angle $\alpha$ (in general it is not absolutely true, but we shall use this assumption in our model), the maximum length of jump is reached when $\alpha=45^\circ$ ,

L=V_0^2/g.

So, if $L=8$ m, we may obtain, that speed with which the bear makes a start from the Earth is approximately equal to

V_0=(L\cdot g)^{1/2}\approx9\,\mathrm{m/s}.

If this speed will be larger than the second cosmic speed for the asteroid, the bear will depart and surely become an independent object of the Kuiper belt. The second cosmic speed may be used for the solution. Nevertheless, the first cosmic speed is better in the situation. Even, if this speed will be less the second cosmic speed, but larger than the first cosmic speed, it would be quite uncomfortable for the bear after it has jumped of the surface: it will follow an orbit of suborbital flight during many hours.

The relationship between the first cosmic speed and the size and density of the asteroid is

V_1^2/R=GM/R^2=G(4/3)\pi R^3\rho/R^2,

V_1^2=G(4/3)\pi R^2\rho=\pi GD^2\rho/3.

The first cosmic speed for an asteroid of the Kuiper belt is equal

V_1=D(\pi G\rho/3)^{1/2}.

D=V_1/(\pi G\rho/3)^{1/2}.

Taking into account that the planned for settling by bears objects of the Kuiper belt consist basically of ice, their characteristic density is about $900\,\mathrm{kg/m^3}$ . So we have that the first cosmic speed $V_1$ is equal to $V_0=(L\cdot g)^{1/2}=9\,\mathrm{m/s}$ for the asteroids with diameter of order

D=V_0/(\pi G\rho/3)^{1/2}=(L\cdot g)^{1/2}/(\pi G\rho/3)^{1/2}=(3Lg/\pi G\rho)^{1/2}.

D\approx35\,\mathrm{km}.

Thus, from the physicists’ point of view it is possible to place polar bears comfortably on ice asteroids with diameters more than about 35 km.

Answer: $D>(3Lg/\pi G\rho)^{1/2}\approx35$ km.

Теми и трудност

Оценена трудност: 4/5.

Механика · Слънчева система · Работа и енергия · Гравитация · Астероиди, комети и метеори

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Оригинал в Архива: betatheo.pdf · официални решения: theosol.pdf · допълнителни данни: IAO2008-data.pdf