| <!DOCTYPE html> | | <!DOCTYPE html> |
| <html lang="en"> | | <html lang="en"> |
| | | |
| <head> | | <head> |
| <meta charset="utf-8"> | | <meta charset="utf-8"> |
| <meta name="viewport" content="width=device-width, initial-scale=1.0"> | | <meta name="viewport" content="width=device-width, initial-scale=1.0"> |
| <meta http-equiv="content-language" content="en"> | | <meta http-equiv="content-language" content="en"> |
| <meta name="keywords" content="Savchenko Problems in Physics, Savchenko solutions, physics problems, physics olympiad preparation, IPhO, Jaan Kalda"> | | <meta name="keywords" content="Savchenko Problems in Physics, Savchenko solutions, physics problems, physics olympiad preparation, IPhO, Jaan Kalda"> |
| <meta name="description" content="A satellite moves around a planet of mass M in an ellipse with semi-major and minor axes a and b. Determine the area that the radius vector drawn from the center of the planet to the satellite "sweeps" per unit time. Find the period of rotation of the satellite."> | | <meta name="description" content="A satellite moves around a planet of mass M in an ellipse with semi-major and minor axes a and b. Determine the area that the radius vector drawn from the center of the planet to the satellite "sweeps" per unit time. Find the period of rotation of the satellite."> |
| <meta name="author" content="Aliaksandr Melnichenka"> | | <meta name="author" content="Aliaksandr Melnichenka"> |
| <meta name="date" content="2023-10" scheme="YYYY-MM"> | | <meta name="date" content="2023-10" scheme="YYYY-MM"> |
| <meta property="og:title" content="A satellite moves around a planet of mass M in an ellipse with semi-major and minor axes a and b. Determine the area that the radius vector drawn from the center of the planet to the satellite "sweeps" per unit time. Find the period of rotation of the satellite."> | | <meta property="og:title" content="A satellite moves around a planet of mass M in an ellipse with semi-major and minor axes a and b. Determine the area that the radius vector drawn from the center of the planet to the satellite "sweeps" per unit time. Find the period of rotation of the satellite."> |
| <meta property="og:image" content="img/logo.png"> | | <meta property="og:image" content="img/logo.png"> |
| <meta property="og:description" content="A satellite moves around a planet of mass M in an ellipse with semi-major and minor axes a and b. Determine the area that the radius vector drawn from the center of the planet to the satellite "sweeps" per unit time. Find the period of rotation of the satellite."> | | <meta property="og:description" content="A satellite moves around a planet of mass M in an ellipse with semi-major and minor axes a and b. Determine the area that the radius vector drawn from the center of the planet to the satellite "sweeps" per unit time. Find the period of rotation of the satellite."> |
| <meta name="yandex-verification" content="6cfda41f74038368"> | | <meta name="yandex-verification" content="6cfda41f74038368"> |
| <title>A satellite moves around a planet of mass M in an ellipse with semi-major and minor axes a and b. Determine the area that the radius vector drawn from the center of the planet to the satellite "sweeps" per unit time. Find the period of rotation of the satellite.</title> | | <title>A satellite moves around a planet of mass M in an ellipse with semi-major and minor axes a and b. Determine the area that the radius vector drawn from the center of the planet to the satellite "sweeps" per unit time. Find the period of rotation of the satellite.</title> |
| <link rel="stylesheet" href="../../css/css-latex/style.css"> | | <link rel="stylesheet" href="../../css/css-latex/style.css"> |
| <link rel="icon" href="../../img/logo.png" type="image/png"> | | <link rel="icon" href="../../img/logo.png" type="image/png"> |
| <script src="../../js/jquery-1.10.1.min.js"></script> | | <script src="../../js/jquery-1.10.1.min.js"></script> |
| <script async src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.7/MathJax.js?config=TeX-MML-AM_CHTML"></script> | | <script async src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.7/MathJax.js?config=TeX-MML-AM_CHTML"></script> |
| <script type="text/x-mathjax-config"> | | <script type="text/x-mathjax-config"> |
| MathJax.Hub.Config({ | | MathJax.Hub.Config({ |
| extensions: ['tex2jax.js'], | | extensions: ['tex2jax.js'], |
| jax: ['input/TeX', 'output/HTML-CSS'], | | jax: ['input/TeX', 'output/HTML-CSS'], |
| tex2jax: { | | tex2jax: { |
| inlineMath: [['$', '$'], ['$', '$']], | | inlineMath: [['$', '$'], ['$', '$']], |
| processEscapes: true, | | processEscapes: true, |
| processClass: 'tex2jax', | | processClass: 'tex2jax', |
| ignoreClass: 'html' | | ignoreClass: 'html' |
| }, | | }, |
| showProcessingMessages: false, | | showProcessingMessages: false, |
| messageStyle: 'none' | | messageStyle: 'none' |
| }); | | }); |
| </script> | | </script> |
| | | |
| <h3> Statement </h3> | | <h3> Statement </h3> |
| <p> | | <p> |
| $2.6.46^*.$ A satellite moves around a planet of mass $M$ in an ellipse with semi-major and minor axes $a$ and $b$. Determine the area that the radius vector drawn from the center of the planet to the satellite "sweeps" per unit time. Find the period of rotation of the satellite. | | $2.6.46^*.$ A satellite moves around a planet of mass $M$ in an ellipse with semi-major and minor axes $a$ and $b$. Determine the area that the radius vector drawn from the center of the planet to the satellite "sweeps" per unit time. Find the period of rotation of the satellite. |
| </p> | | </p> |
| <center> | | <center> |
| <figure> | | <figure> |
| <img src="2.6.46.png" | | <img src="2.6.46.png" |
| loading="lazy" width="230" /> | | loading="lazy" width="230" /> |
| <figcaption> | | <figcaption> |
| For problem $2.6.46^*$ | | For problem $2.6.46^*$ |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| <p> | | <p> |
| </p> | | </p> |
| | | |
| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
| Kepler's Third Law | | Kepler's Third Law |
| $$\frac{T^2}{a^3}=\frac{4\pi ^2}{GM}$$ | | $$\frac{T^2}{a^3}=\frac{4\pi ^2}{GM}$$ |
| From where | | From where |
| $$\boxed{T=2\pi\sqrt{\frac{a^3}{GM}}}$$ | | $$\boxed{T=2\pi\sqrt{\frac{a^3}{GM}}}$$ |
| | | |
| </p> | | </p> |
| <center> | | <center> |
| <figure> | | <figure> |
| <img src="2.6.46_1.png" | | <img src="2.6.46_1.png" |
| loading="lazy" width="290" /> | | loading="lazy" width="290" /> |
| <figcaption> | | <figcaption> |
| $v$ - sweep speed | | $v$ - sweep speed |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| <p> | | <p> |
| Kepler's Second Law | | Kepler's Second Law |
| $$\frac{dS}{dt}=\text{const};\quad \frac{dS}{dt}=v$$ | | $$\frac{dS}{dt}=\text{const};\quad \frac{dS}{dt}=v$$ |
| $$v=\frac{S}{T}=\frac{\pi ab}{2\pi}\cdot\sqrt{\frac{GM}{a^3}}$$ | | $$v=\frac{S}{T}=\frac{\pi ab}{2\pi}\cdot\sqrt{\frac{GM}{a^3}}$$ |
| From here the velocity $v$ could be found as | | From here the velocity $v$ could be found as |
| $$\boxed{v=\frac{1}{2}b\cdot\sqrt{\frac{GM}{a}}}$$ | | $$\boxed{v=\frac{1}{2}b\cdot\sqrt{\frac{GM}{a}}}$$ |
| </p> | | </p> |
| <p style="text-align: right; font-style: italic; font-size: 14;"> | | <p style="text-align: right; font-style: italic; font-size: 14;"> |
| Almaskhan Arsen<br> | | Almaskhan Arsen<br> |
| </p> | | </p> |
| | | |
| | | |
| <h3>Alternative solution</h3> | | <h3>Alternative solution</h3> |
| <p> | | <p> |
| The radius of curvature of the orbit at the apex of the major axis of the ellipse | | The radius of curvature of the orbit at the apex of the major axis of the ellipse |
| \[ | | \[ |
| R = \frac{a}{k^2} = \frac{b^2}{a}. | | R = \frac{a}{k^2} = \frac{b^2}{a}. |
| \] | | \] |
| Therefore | | Therefore |
| \[ | | \[ |
| \frac{v^2}{R} = \frac{v^2 a}{b^2} = \frac{G M}{r^2} \rightarrow vr = \sqrt{G M \frac{b^2}{a}}, | | \frac{v^2}{R} = \frac{v^2 a}{b^2} = \frac{G M}{r^2} \rightarrow vr = \sqrt{G M \frac{b^2}{a}}, |
| \] | | \] |
| \[ | | \[ |
| \frac{dS}{dt} = \frac{1}{2}vr = \frac{1}{2}b \sqrt{\frac{G M}{a}}. | | \frac{dS}{dt} = \frac{1}{2}vr = \frac{1}{2}b \sqrt{\frac{G M}{a}}. |
| \] | | \] |
| Satellite orbital period | | Satellite orbital period |
| \[ | | \[ |
| T = 2\pi \frac{ab}{\frac{dS}{dt}} = 2\pi \frac{a^{3/2}}{\sqrt{G M}}. | | T = 2\pi \frac{ab}{\frac{dS}{dt}} = 2\pi \frac{a^{3/2}}{\sqrt{G M}}. |
| \] | | \] |
| </p> | | </p> |
| | | |
| <h4>Answer</h4> | | <h4>Answer</h4> |
| <p> | | <p> |
| $$T=2\pi\sqrt{\frac{a^3}{GM}}$$ | | $$T=2\pi\sqrt{\frac{a^3}{GM}}$$ |
| $$v=\frac{1}{2}b\cdot\sqrt{\frac{GM}{a}}$$ | | $$v=\frac{1}{2}b\cdot\sqrt{\frac{GM}{a}}$$ |
| </p> | | </p> |
| | | |
| | | |
| <footer class="row container"> | | <footer class="row container"> |
| <br> | | <br> |
| <p> | | <p> |
| <small> © <strong>Savchenko Solutions</strong>, 2023-2024 <br></small> | | <small> © <strong>Savchenko Solutions</strong>, 2023-2024 <br></small> |
| </p> | | </p> |
| <p> | | <p> |
| <small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> alex@savchenkosolutions.com <br></small> | | <small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> alex@savchenkosolutions.com <br></small> |
| </p> | | </p> |
| </footer> | | </footer> |
| </body> | | </body> |
| | | |
| </html> | | </html> |
