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.

Solutions of Savchenko Problems in Physics

Aliaksandr Melnichenka
October 2023

    <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 $r_p$.

    <h3>Solution</h3>
    <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 $R_n$.

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
$$\boxed{dv=\frac{GM\cdot d\varphi}{R_nv_n}}$$

    <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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