To determine how far away the satellite is in its orbit around the Moon when it is at a velocity suitable for a stable orbit, we need to consider the nature of orbital mechanics, which depends on the gravitational attraction of the body being orbited.
The satellite is designed to orbit the Earth at 10,000 km above the surface. This altitude means that in Earth orbit, the total distance from the center of the Earth to the satellite is the radius of the Earth (approximately 6,371 km) plus 10,000 km, which gives a total distance of about 16,371 km from the center of the Earth.
When the satellite is placed in orbit around the Moon, we have to account for the Moon's radius and gravitational force. The radius of the Moon is approximately 1,737 km. If we want the satellite to be orbiting the Moon at the same altitude (10,000 km above the lunar surface), we must add the radius of the Moon to the altitude:
Total distance from the center of the Moon = radius of the Moon + altitude of the satellite = 1,737 km + 10,000 km = 11,737 km
Thus, the answer to the question of how far the satellite is from the center of the Moon in its orbit is:
11,737 km.
However, regarding the specific options provided in your question, we can analyze them:
- Equal to 10,000 km: Incorrect, as this does not account for the Moon's radius.
- Depends on the mass of the satellite: Incorrect, as orbital distance is determined by gravitational forces and altitudes, not the mass of the satellite.
- Farther than 10,000 km: Correct, since the total distance from the center of the Moon is greater than just the 10,000 km.
- Closer than 10,000 km: Incorrect, since it is farther than that.
Therefore, the correct answer would be: farther than 10,000 km.
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