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What are the key differences between circular and elliptical orbits in terms of energy?
Circular: Total, kinetic, potential energy, and angular momentum are constant. Elliptical: Total energy and angular momentum are constant, but kinetic and potential energy vary.
Compare kinetic energy at periapsis and apoapsis in an elliptical orbit.
Periapsis: Kinetic energy is at its maximum. Apoapsis: Kinetic energy is at its minimum.
Compare potential energy at periapsis and apoapsis in an elliptical orbit.
Periapsis: Gravitational potential energy is most negative. Apoapsis: Gravitational potential energy is least negative.
Compare the total mechanical energy required for a satellite to be in a circular orbit versus the energy required to achieve escape velocity.
Circular Orbit: Total energy is negative. Escape Velocity: Total energy is zero.
Compare the motion of the central object and the satellite.
Central Object: Treated as stationary due to its significantly larger mass. Satellite: Orbits around the central object.
What are the steps to derive the escape velocity formula?
1. Start with the total mechanical energy equation: $E = \frac{1}{2}mv^2 - \frac{GMm}{r}$. 2. Set the total energy to zero for escape: $0 = \frac{1}{2}mv_{esc}^2 - \frac{GMm}{r}$. 3. Solve for $v_{esc}$: $v_{esc} = \sqrt{\frac{2GM}{r}}$.
What is the effect of increasing the distance from the central object on gravitational potential energy?
Gravitational potential energy increases (becomes less negative).
What happens to a satellite's kinetic energy as it moves from apoapsis to periapsis in an elliptical orbit?
Kinetic energy increases.
What is the effect of achieving escape velocity?
The object escapes the gravitational pull of the central object and moves away indefinitely.