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en/2.6.45.md
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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 segment connecting the pericenter and apocenter of an elliptical orbit is called the major axis. The ellipse is symmetric relative to it. The segment connecting the points of the orbit farthest from the major axis is called the minor axis. It is perpendicular to the major axis and is also the axis of symmetry of the ellipse. Using the issue conditions 2.6.43, find the probe velocity at the vertices of the minor axis. Express this velocity in terms of the length of the semimajor axis a and the mass of the planet M."> | ||
| + | <meta name="author" content="Aliaksandr Melnichenka"> | ||
| + | <meta name="date" content="2023-10" scheme="YYYY-MM"> | ||
| + | <meta property="og:title" content="The segment connecting the pericenter and apocenter of an elliptical orbit is called the major axis. The ellipse is symmetric relative to it. The segment connecting the points of the orbit farthest from the major axis is called the minor axis. It is perpendicular to the major axis and is also the axis of symmetry of the ellipse. Using the issue conditions, find the probe velocity at the vertices of the minor axis. Express this velocity in terms of the length of the semimajor axis a and the mass of the planet M."> | ||
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| + | <meta property="og:description" content="The segment connecting the pericenter and apocenter of an elliptical orbit is called the major axis. The ellipse is symmetric relative to it. The segment connecting the points of the orbit farthest from the major axis is called the minor axis. It is perpendicular to the major axis and is also the axis of symmetry of the ellipse. Using the issue conditions 2.6.43, find the probe velocity at the vertices of the minor axis. Express this velocity in terms of the length of the semimajor axis a and the mass of the planet M."> | ||
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| + | <title>The segment connecting the pericenter and apocenter of an elliptical orbit is called the major axis. The ellipse is symmetric relative to it. The segment connecting the points of the orbit farthest from the major axis is called the minor axis. It is perpendicular to the major axis and is also the axis of symmetry of the ellipse. Using the issue conditions 2.6.43, find the probe velocity at the vertices of the minor axis. Express this velocity in terms of the length of the semimajor axis a and the mass of the planet M.</title> | ||
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| + | <header style="text-align:center;"> | ||
| + | <div id = "logo"> | ||
| + | <span><img src = "../../img/book.png"><span><span>Savchenko Solutions</span> | ||
| + | </div> | ||
| + | <p class="author"> | ||
| + | Solutions of Savchenko Problems in Physics <br> | ||
| + | <i><b>knowledge must be free</b></i> | ||
| + | </p> | ||
| + | </header> | ||
| + | |||
| + | <h3 id="back-link"><a href="../../#2.6">$\leftarrow$Back</a></h3> | ||
| + | |||
| + | <h3> Statement </h3> | ||
| + | <p> | ||
| + | $2.6.45^*.$ The segment connecting the pericenter and apocenter of an elliptical orbit is called the major axis. The ellipse is symmetric relative to it. The segment connecting the points of the orbit farthest from the major axis is called the minor axis. It is perpendicular to the major axis and is also the axis of symmetry of the ellipse. Using the issue conditions <a href="../2.6.43">2.6.43</a>, find the probe velocity at the vertices of the minor axis. Express this velocity in terms of the length of the semimajor axis a and the mass of the planet $M$. | ||
| + | </p> | ||
| + | |||
| + | <h3>Solution</h3> | ||
| + | <p> | ||
| + | </p> | ||
| + | <center> | ||
| + | <figure> | ||
| + | <img src="2.6.45_1.png" | ||
| + | loading="lazy" width="280" /> | ||
| + | <figcaption> | ||
| + | Major axis of the ellipse | ||
| + | </figcaption> | ||
| + | </figure> | ||
| + | </center> | ||
| + | <p> | ||
| + | From the conservation of energy | ||
| + | $$E= \frac{mv^2}{2}-\frac{GmM}{a}$$ | ||
| + | $$E=-\frac{GmM}{2a}$$ | ||
| + | Equating both expressions | ||
| + | $$\frac{mv^2}{2}-\frac{GmM}{a}=-\frac{GmM}{2a}$$ | ||
| + | $$\frac{mv^2}{2}=\frac{GmM}{2a}$$ | ||
| + | Whence the required velocity | ||
| + | $$\boxed{v=\sqrt{\frac{GM}{a}}}$$ | ||
| + | </p> | ||
| + | |||
| + | <h4>Answer</h4> | ||
| + | <p> | ||
| + | This velocity is parallel to the major axis and perpendicular to the vector $V$, therefore $V_0 = \sqrt{u^2 - V^2}$. Since | ||
| + | |||
| + | \[ | ||
| + | a = \frac{1}{2} \left( r_a + r_p \right) = \frac{p u^2}{u^2 - V^2}, | ||
| + | \] | ||
| + | |||
| + | and | ||
| + | |||
| + | $a = \frac{G M}{V_0^2}$(from the equation $ u^2 / p = G M / p^2 $ for a circular orbit, it follows that $ p u^2 = G M $, | ||
| + | finally | ||
| + | |||
| + | \[ | ||
| + | V_0 = \sqrt{\frac{G M}{a}}. | ||
| + | \] | ||
| + | |||
| + | </p> | ||
| + | <p style="text-align: right; font-style: italic; font-size: 14;"> | ||
| + | Almaskhan Arsen<br> | ||
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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 segment connecting the pericenter and apocenter of an elliptical orbit is called the major axis. The ellipse is symmetric relative to it. The segment connecting the points of the orbit farthest from the major axis is called the minor axis. It is perpendicular to the major axis and is also the axis of symmetry of the ellipse. Using the issue conditions, find the probe velocity at the vertices of the minor axis. Express this velocity in terms of the length of the semimajor axis a and the mass of the planet M."> | |||
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| <span><img src = "../../img/book.png"><span><span>Savchenko Solutions</span> | |||
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| <h3 id="back-link"><a href="../../#2.6">$\leftarrow$Back</a></h3> | |||
| <h3> Statement </h3> | |||
| <p> | |||
| $2.6.45^*.$ The segment connecting the pericenter and apocenter of an elliptical orbit is called the major axis. The ellipse is symmetric relative to it. The segment connecting the points of the orbit farthest from the major axis is called the minor axis. It is perpendicular to the major axis and is also the axis of symmetry of the ellipse. Using the issue conditions <a href="../2.6.43">2.6.43</a>, find the probe velocity at the vertices of the minor axis. Express this velocity in terms of the length of the semimajor axis a and the mass of the planet $M$. | |||
| </p> | |||
| <h3>Solution</h3> | |||
| <p> | |||
| </p> | |||
| <center> | |||
| <figure> | |||
| <img src="2.6.45_1.png" | |||
| loading="lazy" width="280" /> | |||
| <figcaption> | |||
| Major axis of the ellipse | |||
| </figcaption> | |||
| </figure> | |||
| </center> | |||
| <p> | |||
| From the conservation of energy | |||
| $$E= \frac{mv^2}{2}-\frac{GmM}{a}$$ | |||
| $$E=-\frac{GmM}{2a}$$ | |||
| Equating both expressions | |||
| $$\frac{mv^2}{2}-\frac{GmM}{a}=-\frac{GmM}{2a}$$ | |||
| $$\frac{mv^2}{2}=\frac{GmM}{2a}$$ | |||
| Whence the required velocity | |||
| $$\boxed{v=\sqrt{\frac{GM}{a}}}$$ | |||
| </p> | |||
| <h4>Answer</h4> | |||
| <p> | |||
| This velocity is parallel to the major axis and perpendicular to the vector $V$, therefore $V_0 = \sqrt{u^2 - V^2}$. Since | |||
| \[ | |||
| a = \frac{1}{2} \left( r_a + r_p \right) = \frac{p u^2}{u^2 - V^2}, | |||
| \] | |||
| and | |||
| $a = \frac{G M}{V_0^2}$(from the equation $ u^2 / p = G M / p^2 $ for a circular orbit, it follows that $ p u^2 = G M $, | |||
| finally | |||
| \[ | |||
| V_0 = \sqrt{\frac{G M}{a}}. | |||
| \] | |||
| </p> | |||
| <p style="text-align: right; font-style: italic; font-size: 14;"> | |||
| Almaskhan Arsen<br> | |||
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