In the following problems, except where otherwise indicated,...

In the following problems, except where otherwise indicated, assume that the radius of the earth is $6378 \mathrm{km}$, the earth's mass is $5.976\left(10^{24}\right) \mathrm{kg}$, the mass of the sun is $1.99\left(10^{30}\right) \mathrm{kg}, \quad$ and $\quad$ the $\quad$ gravitational $\quad$ constant $\quad$ is $G=66.73\left(10^{-12}\right) \mathrm{m}^{3} /\left(\mathrm{kg} \cdot \mathrm{s}^{2}\right)$. The rocket is traveling around the earth in free flight along the elliptical orbit. If the rocket has the orbit shown, determine the speed of the rocket when it is at $A$ and at $B$.

In the following problems, except where otherwise indicated, assume that the radius of the earth is $6378 \mathrm{km}$, the earth's mass is $5.976\left(10^{24}\right) \mathrm{kg}$, the mass of the sun is $1.99\left(10^{30}\right) \mathrm{kg}, \quad$ and $\quad$ the $\quad$ gravitational $\quad$ constant $\quad$ is $G=66.73\left(10^{-12}\right) \mathrm{m}^{3} /\left(\mathrm{kg} \cdot \mathrm{s}^{2}\right)$.
The rocket is traveling around the earth in free flight along the elliptical orbit. If the rocket has the orbit shown, determine the speed of the rocket when it is at $A$ and at $B$.

Key Concepts

Elliptical Orbit

An elliptical orbit is a type of closed orbital path where the orbiting object moves along an ellipse with one of its foci occupied by the central body. The object's speed is not constant; it increases when the object is closer to the central mass and decreases when it is farther away. This variation is a consequence of the distribution of gravitational forces along the orbit.

Vis-Viva Equation

The vis-viva equation is a key relationship in orbital mechanics that relates the speed of an orbiting body to its distance from the central body and the semi-major axis of the orbit. It is given by v² = GM(2/r - 1/a), where v is the orbital speed, G is the gravitational constant, M is the mass of the central body, r is the instantaneous distance from the central body, and a is the orbit’s semi-major axis. This equation allows the calculation of speed at any point along an elliptical orbit.

Conservation of Energy

Conservation of energy in orbital mechanics refers to the principle that the total mechanical energy (kinetic plus potential) of an orbiting body remains constant if only conservative forces, such as gravity, are doing work. This principle underpins the analysis of orbital motion, allowing one to relate speed and distance along the orbit.

Conservation of Angular Momentum

Conservation of angular momentum indicates that in the absence of external torques, the angular momentum of an orbiting object remains constant. In orbital motion, this means that as the radius (distance from the central body) decreases, the orbital speed increases, and vice versa. This conservation law is crucial for understanding why an object in an elliptical orbit moves faster at points closer to the central mass and slower when further away.

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