8. Satellite and moon. Which travels faster, the moon or a...

8. Satellite and moon. Which travels faster, the moon or a satellite traveling around the earth at a radius just greater than the radius of the earth? What is the ratio of the speeds in terms of the ratio of the radii? What is the ratio of the periods? From the facts that the moon has a period of about 27 days and a radius of orbit $240,000 \mathrm{mi}$ and that the radius of the earth is $4000 \mathrm{mi}$, find the period of the satellite.

Key Concepts

Orbital Mechanics

Orbital mechanics studies the motion of objects moving under the influence of gravity, such as satellites and moons. It involves understanding how gravitational forces govern the trajectories and speeds of these bodies in space, especially in circular or elliptical orbits.

Gravitational and Centripetal Force Balance

In circular orbits, the gravitational force acting on an orbiting body provides the exact centripetal force required to keep the object in its path. Equating gravitational force to centripetal force leads to the derivation of the orbital velocity formula, which shows that the speed of an object in orbit is dependent on its distance from the central mass.

Orbital Velocity

The orbital velocity of an object in a circular orbit is given by the square root of the gravitational constant times the mass of the central body divided by the orbital radius. This relationship implies that an object closer to the central mass must travel faster than one farther away to maintain a stable orbit.

Kepler's Third Law

Kepler's Third Law states that the square of the orbital period is proportional to the cube of the orbital radius. This law provides a way to compare the orbital periods of different bodies by relating their path sizes to the time it takes to complete an orbit, and it is fundamental in calculating period ratios between satellites and moons.

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