The positions of a satellite in elliptical orbit are indicated. Rank gravitational force from greatest to least. Rank speed from greatest to least. Rank momentum from greatest to least. Rank acceleration from greatest to least.
The Correct Answer and Explanation is:
In the case of a satellite moving in an elliptical orbit, the various quantities—gravitational force, speed, momentum, and acceleration—change depending on the satellite’s position in the orbit. Let’s analyze the behavior of each quantity at different points of the orbit.
1. Gravitational Force
Gravitational force is governed by the law of universal gravitation, which states that F=GMmr2F = \frac{GMm}{r^2}F=r2GMm, where GGG is the gravitational constant, MMM is the mass of the central body (like a planet), mmm is the mass of the satellite, and rrr is the distance between the satellite and the center of the central body. Since gravitational force is inversely proportional to the square of the distance, the force will be greatest when the satellite is closest to the central body (periapsis) and least when it is farthest away (apoapsis).
Ranking of gravitational force (greatest to least):
- Periapsis (closest to the central body)
- Apoapsis (farthest from the central body)
2. Speed
The speed of a satellite in an elliptical orbit is highest at periapsis and lowest at apoapsis. This behavior follows from the conservation of angular momentum in an elliptical orbit, meaning that when the satellite is closer to the central body, it must move faster to conserve angular momentum.
Ranking of speed (greatest to least):
- Periapsis
- Apoapsis
3. Momentum
Momentum is the product of the satellite’s mass and velocity: p=mvp = mvp=mv. Since mass does not change, momentum will follow the same pattern as speed. Therefore, the satellite will have the greatest momentum at periapsis and the least momentum at apoapsis.
Ranking of momentum (greatest to least):
- Periapsis
- Apoapsis
4. Acceleration
The acceleration experienced by a satellite in orbit is primarily due to the gravitational force exerted by the central body, which changes with the satellite’s distance. The acceleration is greatest at periapsis, where the satellite is closest to the central body, and it is least at apoapsis. This can be deduced from the fact that gravitational force, and thus acceleration, follows an inverse square law.
Ranking of acceleration (greatest to least):
- Periapsis
- Apoapsis
Conclusion:
- Gravitational Force: Periapsis > Apoapsis
- Speed: Periapsis > Apoapsis
- Momentum: Periapsis > Apoapsis
- Acceleration: Periapsis > Apoapsis
This ranking reflects the elliptical nature of the orbit and how the satellite’s motion is influenced by its changing distance from the central body.
