<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 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">
@@ -14,9 +14,9 @@
<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.">
<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>
$2.6.42^*.$ 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
Kepler's second law states that the speed of sweeping out a sector area is constant, i.e.
$$
\frac{dS}{dt}=\text{const}
$$
We know the satellite's distance $r_n$ at the pericenter and its velocity $v_n$ at this point. At the pericenter, over a small interval $dt$, a very small sector is swept out, which can be considered an isosceles triangle with the thigh length $r_n$ and the base $v_n dt$. So at this point $dS_n=\frac{1}{2}v_ndt \cdot r_n$, or
$$
\frac{dS}{dt}=\frac{v_nr_n}{2}.
$$
Now let's move on to the question of the change in speed with each passage of the angle $d\varphi$. If at the moment the distance between the satellite and the planet is equal to $r$, then
where the index $r$ denotes the change in the radial component of the velocity in the direction toward the center of the planet. From here we obtain the final answer:
<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>
<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="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="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 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: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:image" content="img/logo.png">
@@ -14,9 +14,9 @@
<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.">
<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.">
<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>
<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>
$2.6.42^*.$ 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 $r_p$.
$2.6.42^*.$ 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>
</p>
<center>
<center>
<figure>
<figure>
<img src="2.6.42.png"
<img src="2.6.42.png"
loading="lazy" width="120" />
loading="lazy" width="120" />
<figcaption>
<figcaption>
For problem $2.6.42^*$
For problem $2.6.42^*$
</figcaption>
</figcaption>
</figure>
</figure>
</center>
</center>
<p>
<p>
</p>
</p>
<h3>Solution</h3>
<h3>Solution</h3>
<p>
<p>
</p>
</p>
<center>
<center>
<figure>
<figure>
<img src="2.6.42_1.png"
<img src="2.6.42_1.png"
loading="lazy" width="300" />
loading="lazy" width="300" />
<figcaption>
<figcaption>
Change in speed over a short period of time
Change in speed over a short period of time
</figcaption>
</figcaption>
</figure>
</figure>
</center>
</center>
<p>
<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>
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
Newton's second law for an external force, the gravitational force
Kepler's second law states that the speed of sweeping out a sector area is constant, i.e.
Kepler's second law states that the speed of sweeping out a sector area is constant, i.e.
$$
$$
\frac{dS}{dt}=\text{const}
\frac{dS}{dt}=\text{const}
$$
$$
We know the satellite's distance $r_n$ at the pericenter and its velocity $v_n$ at this point. At the pericenter, over a small interval $dt$, a very small sector is swept out, which can be considered an isosceles triangle with the thigh length $r_n$ and the base $v_n dt$. So at this point $dS_n=\frac{1}{2}v_ndt \cdot r_n$, or
We know the satellite's distance $r_n$ at the pericenter and its velocity $v_n$ at this point. At the pericenter, over a small interval $dt$, a very small sector is swept out, which can be considered an isosceles triangle with the thigh length $r_n$ and the base $v_n dt$. So at this point $dS_n=\frac{1}{2}v_ndt \cdot r_n$, or
$$
$$
\frac{dS}{dt}=\frac{v_nr_n}{2}.
\frac{dS}{dt}=\frac{v_nr_n}{2}.
$$
$$
Now let's move on to the question of the change in speed with each passage of the angle $d\varphi$. If at the moment the distance between the satellite and the planet is equal to $r$, then
Now let's move on to the question of the change in speed with each passage of the angle $d\varphi$. If at the moment the distance between the satellite and the planet is equal to $r$, then
where the index $r$ denotes the change in the radial component of the velocity in the direction toward the center of the planet. From here we obtain the final answer:
where the index $r$ denotes the change in the radial component of the velocity in the direction toward the center of the planet. From here we obtain the final answer:
<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>
