| <!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="Two probes are launched from an orbital station moving at a speed of u in a circular orbit around the planet. The initial velocity of the probes relative to the planet is v (\sqrt{2} u > v > u). One probe moves in the direction of the planet's radius; the initial velocity of the other probe is perpendicular to its radius. Find the ratio of the maximum possible distances from the probes to the center of the planet."> | | <meta name="description" content="Two probes are launched from an orbital station moving at a speed of u in a circular orbit around the planet. The initial velocity of the probes relative to the planet is v (\sqrt{2} u > v > u). One probe moves in the direction of the planet's radius; the initial velocity of the other probe is perpendicular to its radius. Find the ratio of the maximum possible distances from the probes to the center of the planet."> |
| <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="Two probes are launched from an orbital station moving at a speed of u in a circular orbit around the planet. The initial velocity of the probes relative to the planet is v (\sqrt{2} u > v > u). One probe moves in the direction of the planet's radius; the initial velocity of the other probe is perpendicular to its radius. Find the ratio of the maximum possible distances from the probes to the center of the planet."> | | <meta property="og:title" content="Two probes are launched from an orbital station moving at a speed of u in a circular orbit around the planet. The initial velocity of the probes relative to the planet is v (\sqrt{2} u > v > u). One probe moves in the direction of the planet's radius; the initial velocity of the other probe is perpendicular to its radius. Find the ratio of the maximum possible distances from the probes to the center of the planet."> |
| <meta property="og:image" content="img/logo.png"> | | <meta property="og:image" content="img/logo.png"> |
| <meta property="og:description" content="Two probes are launched from an orbital station moving at a speed of u in a circular orbit around the planet. The initial velocity of the probes relative to the planet is v (\sqrt{2} u > v > u). One probe moves in the direction of the planet's radius; the initial velocity of the other probe is perpendicular to its radius. Find the ratio of the maximum possible distances from the probes to the center of the planet."> | | <meta property="og:description" content="Two probes are launched from an orbital station moving at a speed of u in a circular orbit around the planet. The initial velocity of the probes relative to the planet is v (\sqrt{2} u > v > u). One probe moves in the direction of the planet's radius; the initial velocity of the other probe is perpendicular to its radius. Find the ratio of the maximum possible distances from the probes to the center of the planet."> |
| <meta name="yandex-verification" content="6cfda41f74038368"> | | <meta name="yandex-verification" content="6cfda41f74038368"> |
| <title>Two probes are launched from an orbital station moving at a speed of u in a circular orbit around the planet. The initial velocity of the probes relative to the planet is v (\sqrt{2} u > v > u). One probe moves in the direction of the planet's radius; the initial velocity of the other probe is perpendicular to its radius. Find the ratio of the maximum possible distances from the probes to the center of the planet.</title> | | <title>Two probes are launched from an orbital station moving at a speed of u in a circular orbit around the planet. The initial velocity of the probes relative to the planet is v (\sqrt{2} u > v > u). One probe moves in the direction of the planet's radius; the initial velocity of the other probe is perpendicular to its radius. Find the ratio of the maximum possible distances from the probes to the center of the planet.</title> |
| <link rel="stylesheet" href="https://savchenkosolutions.com/css/css-latex/style.css"> | | <link rel="stylesheet" href="https://savchenkosolutions.com/css/css-latex/style.css"> |
| <link rel="icon" href="https://savchenkosolutions.com/img/logo.png" type="image/png"> | | <link rel="icon" href="https://savchenkosolutions.com/img/logo.png" type="image/png"> |
| <script src="https://savchenkosolutions.com/js/jquery-1.10.1.min.js"></script> | | <script src="https://savchenkosolutions.com/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> |
| </head> | | </head> |
| <body style=""> | | <body style=""> |
| <header style="text-align:center;"> | | <header style="text-align:center;"> |
| <h2>Solutions of Savchenko Problems in Physics</h2> | | <h2>Solutions of Savchenko Problems in Physics</h2> |
| <p class="author"> | | <p class="author"> |
| Aliaksandr Melnichenka <br/> | | Aliaksandr Melnichenka <br/> |
| October 2023 | | October 2023 |
| </p> | | </p> |
| </header> | | </header> |
| | | |
| <h3 id="back-link"><a href="../../#2.6">$\leftarrow$Back</a></h3> | | <h3 id="back-link"><a href="../../#2.6">$\leftarrow$Back</a></h3> |
| | | |
| <h3> Statement </h3> | | <h3> Statement </h3> |
| <p> | | <p> |
| $2.6.41^*.$ Two probes are launched from an orbital station moving at a speed of $u$ in a circular orbit around the planet. The initial velocity of the probes relative to the planet is $v$ ($\sqrt{2} u > v > u$). One probe moves in the direction of the planet's radius; the initial velocity of the other probe is perpendicular to its radius. Find the ratio of the maximum possible distances from the probes to the center of the planet. | | $2.6.41^*.$ Two probes are launched from an orbital station moving at a speed of $u$ in a circular orbit around the planet. The initial velocity of the probes relative to the planet is $v$ ($\sqrt{2} u > v > u$). One probe moves in the direction of the planet's radius; the initial velocity of the other probe is perpendicular to its radius. Find the ratio of the maximum possible distances from the probes to the center of the planet. |
| </p> | | </p> |
| | | |
| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
| Let's consider two cases:</p><p> | | Let's consider two cases:</p><p> |
| 1) Velocity $\vec{v}$ in radial direction:</p><p> | | 1) Velocity $\vec{v}$ in radial direction:</p><p> |
| For this case, let's consider preservation of energy | | For this case, let's consider preservation of energy |
| $$\frac{mv^2}{2}-\frac{GmM}{R_0}=\frac{m{v'}^2}{2}-\frac{GmM}{R_1}$$ | | $$\frac{mv^2}{2}-\frac{GmM}{R_0}=\frac{m{v'}^2}{2}-\frac{GmM}{R_1}$$ |
| Analyzing the components, when $v'\to 0,$ $R_1$ tends to the maximum value $R_1 \to R_\text{1max}$ | | Analyzing the components, when $v'\to 0,$ $R_1$ tends to the maximum value $R_1 \to R_\text{1max}$ |
| $$\frac{mv^2}{2} - \frac{GmM}{R_0} = - \frac{GmM}{R_\text{1max}}$$ | | $$\frac{mv^2}{2} - \frac{GmM}{R_0} = - \frac{GmM}{R_\text{1max}}$$ |
| | | |
| $$\frac{1}{2}v^2 = GM \left( \frac{1}{R_0} - \frac{1}{R_\text{1max}} \right) $$ | | $$\frac{1}{2}v^2 = GM \left( \frac{1}{R_0} - \frac{1}{R_\text{1max}} \right) $$ |
| 2) The velocity $v$ is perpendicular to the radius</p><p> | | 2) The velocity $v$ is perpendicular to the radius</p><p> |
| From Kepler's first law, the orbit will be an ellipse | | From Kepler's first law, the orbit will be an ellipse |
| </p> | | </p> |
| <center> | | <center> |
| <figure> | | <figure> |
| <img src="2.6.41_1.png" | | <img src="2.6.41_1.png" |
| loading="lazy" width="300" /> | | loading="lazy" width="300" /> |
| <figcaption> | | <figcaption> |
| Elliptical orbit | | Elliptical orbit |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| <p> | | <p> |
| <i>$R_0$ - radius of circular orbit<br> | | <i>$R_0$ - radius of circular orbit<br> |
| $R_1$ - maximum orbit at radial velocity<br> | | $R_1$ - maximum orbit at radial velocity<br> |
| $R_2$ - maximum orbit at perpendicular velocity</i></p><p> | | $R_2$ - maximum orbit at perpendicular velocity</i></p><p> |
| | | |
| Similarly, the preservation of energy | | Similarly, the preservation of energy |
| $$\frac{mv^2}{2}-\frac{GmM}{R_0}=\frac{m{v'}^2}{2}-\frac{GmM}{R_\text{2max}}\quad(1)$$ | | $$\frac{mv^2}{2}-\frac{GmM}{R_0}=\frac{m{v'}^2}{2}-\frac{GmM}{R_\text{2max}}\quad(1)$$ |
| Conservation of angular momentum | | Conservation of angular momentum |
| $$mvR_0=m{v'}R_\text{2max}$$ | | $$mvR_0=m{v'}R_\text{2max}$$ |
| From where | | From where |
| $$v'=v\frac{R_0}{R_\text{2max}}\quad(2)$$ | | $$v'=v\frac{R_0}{R_\text{2max}}\quad(2)$$ |
| Substituting $(2)$ into $(1)$ | | Substituting $(2)$ into $(1)$ |
| $$\frac{v^2}{2}\left(1+\frac{R_0}{R_\text{2max}}\right) = GM \frac{1}{R_0}$$ | | $$\frac{v^2}{2}\left(1+\frac{R_0}{R_\text{2max}}\right) = GM \frac{1}{R_0}$$ |
