What are the steps to derive the escape velocity formula?

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 $v_{esc}$ : $v_{esc} = \sqrt{\frac{2GM}{r}}$ .

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What are the steps to derive the escape velocity formula?

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.

Compare circular and elliptical orbits in terms of energy.

Circular: Constant kinetic and potential energy, constant total mechanical energy. Elliptical: Fluctuating kinetic and potential energy, constant total mechanical energy.

Compare the speed of a satellite in circular vs. elliptical orbits.

Circular: Constant speed. Elliptical: Speed varies; highest at periapsis, lowest at apoapsis.