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Консультация № 132477
16.04.2008, 19:10
0.00 руб.
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1
1
Всем здрасьте))
Собственно задачка, буду благодарен за решение)
Искусственный спутник Земли запущен с экватора и движется по круговой орбите в плоскости экватора в направлении вращения Земли. Найти отношение радиуса орбиты спутника к радиусу Земли, при условии, что спутник переодически, раз в двое суток проходит над точкой запуска. Радиус Земли 6400км g=9,8 мс2, T=24часа
Я так понимаю, что она по 3-му закону кеплера решается?
(R1/R2)^3=(T1/T2)^2 ? помогите плс)
Собственно задачка, буду благодарен за решение)
Искусственный спутник Земли запущен с экватора и движется по круговой орбите в плоскости экватора в направлении вращения Земли. Найти отношение радиуса орбиты спутника к радиусу Земли, при условии, что спутник переодически, раз в двое суток проходит над точкой запуска. Радиус Земли 6400км g=9,8 мс2, T=24часа
Я так понимаю, что она по 3-му закону кеплера решается?
(R1/R2)^3=(T1/T2)^2 ? помогите плс)
Обсуждение
19.04.2008, 12:44
общий
это ответ
Здравствуйте, Drezden!
Пожалуй, неплохо по 3-му закону Кеплера (раз требуется найти именно "отношение", а не сам радиус), но для этого сначала надо определить T<sub>1</sub>; это делается, исходя из уравнения: g = V<sub>1к</sub><sup>2</sup>/R<sub>1</sub> (1), где V<sub>1к</sub> - "первая космическая скорость" - в нашем случае скорость, при которой центробежная сила уравновешивает "притяженье Земли" как раз на расстоянии R<sub>1</sub> от центра. Из (1): V<sub>1к</sub> = √(g*R<sub>1</sub>) (2). Период обращения T<sub>1</sub> при этом равен длине окружности, делённой на V<sub>1к</sub>, т.е. T<sub>1</sub> = 2*π*R<sub>1</sub>/V<sub>1к</sub> (3), а с учётом (2): T<sub>1</sub> = 2*π*√(R<sub>1</sub>/g) (4). Тогда: (T<sub>2</sub>/T<sub>1</sub>)<sup>2</sup> = T<sub>2</sub><sup>2</sup>*g/(R<sub>1</sub>*(2*π)<sup>2</sup>) (5). По условию T<sub>2</sub> = 2*24*60*60 = 172800 с; подставив и прочие цифры: (T<sub>2</sub>/T<sub>1</sub>)<sup>2</sup> = 172800<sup>2</sup>*9.8/(6400*1000*(2*π)<sup>2</sup>) = 1158 (6), а по 3-му закону Кеплера: R<sub>2</sub>/R<sub>1</sub> = <sup>3</sup>√((T<sub>2</sub>/T<sub>1</sub>)<sup>2</sup>) = <sup>3</sup>√(1158) = 10.5
Пожалуй, неплохо по 3-му закону Кеплера (раз требуется найти именно "отношение", а не сам радиус), но для этого сначала надо определить T<sub>1</sub>; это делается, исходя из уравнения: g = V<sub>1к</sub><sup>2</sup>/R<sub>1</sub> (1), где V<sub>1к</sub> - "первая космическая скорость" - в нашем случае скорость, при которой центробежная сила уравновешивает "притяженье Земли" как раз на расстоянии R<sub>1</sub> от центра. Из (1): V<sub>1к</sub> = √(g*R<sub>1</sub>) (2). Период обращения T<sub>1</sub> при этом равен длине окружности, делённой на V<sub>1к</sub>, т.е. T<sub>1</sub> = 2*π*R<sub>1</sub>/V<sub>1к</sub> (3), а с учётом (2): T<sub>1</sub> = 2*π*√(R<sub>1</sub>/g) (4). Тогда: (T<sub>2</sub>/T<sub>1</sub>)<sup>2</sup> = T<sub>2</sub><sup>2</sup>*g/(R<sub>1</sub>*(2*π)<sup>2</sup>) (5). По условию T<sub>2</sub> = 2*24*60*60 = 172800 с; подставив и прочие цифры: (T<sub>2</sub>/T<sub>1</sub>)<sup>2</sup> = 172800<sup>2</sup>*9.8/(6400*1000*(2*π)<sup>2</sup>) = 1158 (6), а по 3-му закону Кеплера: R<sub>2</sub>/R<sub>1</sub> = <sup>3</sup>√((T<sub>2</sub>/T<sub>1</sub>)<sup>2</sup>) = <sup>3</sup>√(1158) = 10.5
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