Rank speed from greatest to least at each point. Rank Kinetic Energy from greatest to least at each point. Rank Potential Energy from greatest to least at each point.
The Correct Answer and Explanation is:
To accurately rank speed, kinetic energy, and potential energy at different points in an elliptical orbit, we need to recall a few key principles of orbital mechanics:
1. Speed
In an elliptical orbit, the satellite moves faster when it is closer to the central body (at periapsis) and slower when it is farther from the central body (at apoapsis). This behavior is explained by Kepler’s Second Law, which states that a line joining a planet and the sun sweeps out equal areas during equal intervals of time. To conserve angular momentum, the satellite must move faster when closer to the central body and slower when farther away.
Thus, the ranking of speed from greatest to least would be:
- Greatest Speed: At periapsis (closest point)
- Least Speed: At apoapsis (farthest point)
2. Kinetic Energy
Kinetic energy (KE) is related to speed and is given by the formula KE=12mv2KE = \frac{1}{2} mv^2KE=21mv2, where mmm is the mass and vvv is the speed. Since the speed is greatest at periapsis and least at apoapsis, the kinetic energy is greatest at periapsis and least at apoapsis.
The ranking of kinetic energy from greatest to least would be:
- Greatest Kinetic Energy: At periapsis
- Least Kinetic Energy: At apoapsis
3. Potential Energy
Gravitational potential energy (PE) in orbit is negative and depends on the distance from the central body, given by PE=−GMmrPE = -\frac{GMm}{r}PE=−rGMm, where GGG is the gravitational constant, MMM is the mass of the central body, mmm is the mass of the satellite, and rrr is the distance from the center of the central body. The potential energy is more negative (i.e., less bound) when the satellite is farther away (at apoapsis) and less negative (i.e., more bound) when it is closer (at periapsis).
Thus, the ranking of potential energy from greatest to least would be:
- Least (most negative) Potential Energy: At periapsis
- Greatest Potential Energy: At apoapsis
Summary:
- Speed: Greatest at periapsis, least at apoapsis.
- Kinetic Energy: Greatest at periapsis, least at apoapsis.
- Potential Energy: Greatest at apoapsis, least at periapsis.
These relationships reflect the trade-off between kinetic and potential energy, maintaining the conservation of mechanical energy in the orbiting system.
