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What are the key differences between circular and elliptical orbits?
Circular: Constant speed, gravitational force always perpendicular to velocity. | Elliptical: Varying speed, gravitational force not always perpendicular to velocity.
Compare and contrast Kepler's First and Second Laws.
First Law: Describes the shape of the orbit (ellipse). | Second Law: Describes the speed of the planet at different points in the orbit (equal areas in equal times).
What is the effect of increasing the mass of a central body on the orbital velocity of a satellite?
The orbital velocity of the satellite increases.
What is the effect of increasing the radius of an orbit on the orbital period?
The orbital period increases.
What is the effect of a spacecraft passing close to a planet during a gravity assist?
The spacecraft's speed and/or direction changes due to the interaction with the planet's gravitational field.
What happens if a satellite's initial velocity is too slow?
The satellite falls inward towards the central body.
What happens if a satellite's initial velocity is too fast?
The satellite may escape the orbit.
How do you derive orbital velocity for a circular orbit?
1. Set gravitational force equal to centripetal force: $G rac{Mm}{r^2} = m rac{v^2}{r}$. 2. Solve for $v$: $v = \sqrt{\frac{GM}{r}}$
How do you calculate orbital period for a circular orbit?
1. Use the formula: $T = \frac{2\pi r}{v}$. 2. Substitute the expression for orbital velocity: $T = 2\pi \sqrt{\frac{r^3}{GM}}$
Describe the process of a gravity assist.
1. Spacecraft approaches a moving planet. 2. Spacecraft interacts with the planet's gravitational field. 3. Spacecraft gains or loses kinetic energy, changing its speed and/or direction. 4. Planet's velocity change is negligible.