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 a satellite's distance from the central object on its gravitational potential energy?

The gravitational potential energy becomes less negative (increases).

What happens to a satellite's motion when its total mechanical energy is zero?

The satellite achieves escape velocity and breaks free from the central object's gravity, moving away until its speed is zero at an infinite distance.

What is the effect of a satellite getting closer to the central object?

The gravitational potential energy becomes more negative.

What is the effect of increasing the radius of a circular orbit on the satellite's orbital speed?

The satellite's orbital speed decreases.

What is the effect of a satellite reaching escape velocity?

The satellite breaks free from the central object's gravity and continues moving away until its speed is zero at an infinite distance.

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.