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| + | <meta name="author" content="Aliaksandr Melnichenka"> | ||
| + | <meta name="date" content="2023-10" scheme="YYYY-MM"> | ||
| + | <meta property="og:title" content="The plane of the satellite's orbit is divided into sectors with a common vertex in the center of the planet of mass M and the same small solution angles d \varphi. Find the change in the satellite's speed as it passes through each sector, if its speed is in the pericenter v_p, and the distance from the satellite to the center of the planet is in the pericenter r_p."> | ||
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| + | <title>The plane of the satellite's orbit is divided into sectors with a common vertex in the center of the planet of mass M and the same small solution angles d \varphi. Find the change in the satellite's speed as it passes through each sector, if its speed is in the pericenter v_p, and the distance from the satellite to the center of the planet is in the pericenter r_p.</title> | ||
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| + | <header style="text-align:center;"> | ||
| + | <h2>Solutions of Savchenko Problems in Physics</h2> | ||
| + | <p class="author"> | ||
| + | Aliaksandr Melnichenka <br/> | ||
| + | October 2023 | ||
| + | </p> | ||
| + | </header> | ||
| + | |||
| + | <h3 id="back-link"><a href="../../#2.6">$\leftarrow$Back</a></h3> | ||
| + | |||
| + | <h3> Statement </h3> | ||
| + | <p> | ||
| + | $2.6.42^*.$ The plane of the satellite's orbit is divided into sectors with a common vertex in the center of the planet of mass $M$ and the same small solution angles $d \varphi$. Find the change in the satellite's speed as it passes through each sector, if its speed is in the pericenter $v_p$, and the distance from the satellite to the center of the planet is in the pericenter $r_p$. | ||
| + | </p> | ||
| + | <center> | ||
| + | <figure> | ||
| + | <img src="2.6.42.png" | ||
| + | loading="lazy" width="120" /> | ||
| + | <figcaption> | ||
| + | For problem $2.6.42^*$ | ||
| + | </figcaption> | ||
| + | </figure> | ||
| + | </center> | ||
| + | <p> | ||
| + | </p> | ||
| + | |||
| + | <h3>Solution</h3> | ||
| + | <p> | ||
| + | |||
| + | </p> | ||
| + | <center> | ||
| + | <figure> | ||
| + | <img src="2.6.42_1.png" | ||
| + | loading="lazy" width="300" /> | ||
| + | <figcaption> | ||
| + | Change in speed over a short period of time | ||
| + | </figcaption> | ||
| + | </figure> | ||
| + | </center> | ||
| + | <p> | ||
| + | |||
| + | For a short time, $dt$, the body moves along the arc of the circle of arc $d\varphi$ with radius-curvature of the trajectory $R_n$.</p><p> | ||
| + | Newton's second law for an external force, the gravitational force | ||
| + | $$ma=\frac{GmM}{R_n^2}$$ | ||
| + | By definition of acceleration | ||
| + | $$a=\frac{dv}{dt}\Rightarrow \frac{dv}{dt}=\frac{GM}{R_n^2}\quad (1)$$ | ||
| + | In a small amount of time | ||
| + | $$R_n=v_ndt\quad (2)$$ | ||
| + | By definition the arc length of the circle corresponding to the angle $d\varphi$ | ||
| + | $$dR=d\varphi \cdot R_n\quad (3)$$ | ||
| + | After substituting $(2)$ and $(3)$ into $(1)$ | ||
| + | $$\frac{dv}{dR}v_n=\frac{GM}{R_n^2}\Rightarrow \frac{dv}{d\varphi}v_n=\frac{GM}{R_n}$$ | ||
| + | After the transformation, we can get a change in the satellite's speed | ||
| + | $$\boxed{dv=\frac{GM\cdot d\varphi}{R_nv_n}}$$ | ||
| + | </p> | ||
| + | |||
| + | <h4>Answer</h4> | ||
| + | <p> | ||
| + | $dv=\gamma Md\varphi/(v_{\mathrm{n}}r_{\mathrm{n}})$. The vector $dv$ is directed towards the centre of the planet | ||
| + | </p> | ||
| + | <p style="text-align: right; font-style: italic; font-size: 14;"> | ||
| + | Almaskhan Arsen<br> | ||
| + | </p> | ||
| + | |||
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| <meta name="keywords" content="Savchenko Problems in Physics, Savchenko solutions, physics problems, physics olympiad preparation, IPhO, Jaan Kalda"> | |||
| <meta name="description" content="The plane of the satellite's orbit is divided into sectors with a common vertex in the center of the planet of mass M and the same small solution angles d \varphi. Find the change in the satellite's speed as it passes through each sector, if its speed is in the pericenter v_p, and the distance from the satellite to the center of the planet is in the pericenter r_p."> | |||
| <meta name="author" content="Aliaksandr Melnichenka"> | |||
| <meta name="date" content="2023-10" scheme="YYYY-MM"> | |||
| <meta property="og:title" content="The plane of the satellite's orbit is divided into sectors with a common vertex in the center of the planet of mass M and the same small solution angles d \varphi. Find the change in the satellite's speed as it passes through each sector, if its speed is in the pericenter v_p, and the distance from the satellite to the center of the planet is in the pericenter r_p."> | |||
| <meta property="og:image" content="img/logo.png"> | |||
| <meta property="og:description" content="The plane of the satellite's orbit is divided into sectors with a common vertex in the center of the planet of mass M and the same small solution angles d \varphi. Find the change in the satellite's speed as it passes through each sector, if its speed is in the pericenter v_p, and the distance from the satellite to the center of the planet is in the pericenter r_p."> | |||
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| <title>The plane of the satellite's orbit is divided into sectors with a common vertex in the center of the planet of mass M and the same small solution angles d \varphi. Find the change in the satellite's speed as it passes through each sector, if its speed is in the pericenter v_p, and the distance from the satellite to the center of the planet is in the pericenter r_p.</title> | |||
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| </head> | |||
| <body style=""> | |||
| <header style="text-align:center;"> | |||
| <h2>Solutions of Savchenko Problems in Physics</h2> | |||
| <p class="author"> | |||
| Aliaksandr Melnichenka <br/> | |||
| October 2023 | |||
| </p> | |||
| </header> | |||
| <h3 id="back-link"><a href="../../#2.6">$\leftarrow$Back</a></h3> | |||
| <h3> Statement </h3> | |||
| <p> | |||
| $2.6.42^*.$ The plane of the satellite's orbit is divided into sectors with a common vertex in the center of the planet of mass $M$ and the same small solution angles $d \varphi$. Find the change in the satellite's speed as it passes through each sector, if its speed is in the pericenter $v_p$, and the distance from the satellite to the center of the planet is in the pericenter $r_p$. | |||
| </p> | |||
| <center> | |||
| <figure> | |||
| <img src="2.6.42.png" | |||
| loading="lazy" width="120" /> | |||
| <figcaption> | |||
| For problem $2.6.42^*$ | |||
| </figcaption> | |||
| </figure> | |||
| </center> | |||
| <p> | |||
| </p> | |||
| <h3>Solution</h3> | |||
| <p> | |||
| </p> | |||
| <center> | |||
| <figure> | |||
| <img src="2.6.42_1.png" | |||
| loading="lazy" width="300" /> | |||
| <figcaption> | |||
| Change in speed over a short period of time | |||
| </figcaption> | |||
| </figure> | |||
| </center> | |||
| <p> | |||
| For a short time, $dt$, the body moves along the arc of the circle of arc $d\varphi$ with radius-curvature of the trajectory $R_n$.</p><p> | |||
| Newton's second law for an external force, the gravitational force | |||
| $$ma=\frac{GmM}{R_n^2}$$ | |||
| By definition of acceleration | |||
| $$a=\frac{dv}{dt}\Rightarrow \frac{dv}{dt}=\frac{GM}{R_n^2}\quad (1)$$ | |||
| In a small amount of time | |||
| $$R_n=v_ndt\quad (2)$$ | |||
| By definition the arc length of the circle corresponding to the angle $d\varphi$ | |||
| $$dR=d\varphi \cdot R_n\quad (3)$$ | |||
| After substituting $(2)$ and $(3)$ into $(1)$ | |||
| $$\frac{dv}{dR}v_n=\frac{GM}{R_n^2}\Rightarrow \frac{dv}{d\varphi}v_n=\frac{GM}{R_n}$$ | |||
| After the transformation, we can get a change in the satellite's speed | |||
| $$\boxed{dv=\frac{GM\cdot d\varphi}{R_nv_n}}$$ | |||
| </p> | |||
| <h4>Answer</h4> | |||
| <p> | |||
| $dv=\gamma Md\varphi/(v_{\mathrm{n}}r_{\mathrm{n}})$. The vector $dv$ is directed towards the centre of the planet | |||
| </p> | |||
| <p style="text-align: right; font-style: italic; font-size: 14;"> | |||
| Almaskhan Arsen<br> | |||
| </p> | |||
| <footer class="row container"> | |||
| <br> | |||
| <p> | |||
| <small> © <strong>Savchenko Solutions</strong>, 2023-2024 <br></small> | |||
| </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> | |||
| </p> | |||
| </footer> | |||
| </body> | |||
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