A satellite orbiting a planet experiences only the gravitational force, which acts along the line connecting the satellite to the planet's center. This central force produces no torque about the planet's center, so angular momentum is conserved.

Additionally, gravity is a conservative force, meaning mechanical energy is conserved throughout the orbit. These two conservation laws together provide powerful tools for analyzing orbital motion.

The Conservation Equations

For a satellite of mass $m$ orbiting a planet of mass $M$, the total mechanical energy is the sum of kinetic and gravitational potential energy:

This quantity remains constant throughout the orbit. The angular momentum about the planet's center is given by $L = mvr\sin\theta$, where $\theta$ is the angle between $\vec{v}$ and $\vec{r}$.

At special points called periapsis (closest approach) and apoapsis (farthest distance), the velocity is perpendicular to the radial direction, so $\sin\theta = 1$ and the angular momentum simplifies to:

Relating Speeds at Different Orbital Positions

Consider a satellite at periapsis with distance $r_p$ and speed $v_p$, and at apoapsis with distance $r_a$ and speed $v_a$. Conservation of angular momentum gives:

which simplifies to $v_p r_p = v_a r_a$. Conservation of energy provides a second equation:

With two equations and two unknowns, you can solve for the speed at one position given the speed and distance at another position, along with the second distance.

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