➡The Moon's mass is 7.4 x 10²2 kg, and its radius is 1700 km. What would be the period and the speed of a spacecraft moving in a circular orbit just above the lunar surface?
➡The Moon's mass is 7.4 x 10²2 kg, and its radius is 1700 km. What would be the period and the speed of a spacecraft moving in a circular orbit just above the lunar surface?
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![### Orbital Mechanics of a Spacecraft around the Moon
**Problem Statement:**
The Moon's mass is \(7.4 \times 10^{22}\) kg, and its radius is 1700 km. What would be the period and the speed of a spacecraft moving in a circular orbit just above the lunar surface?
**Solution Approach:**
To solve this problem, we need to consider the fundamental principles of orbital mechanics.
1. **Orbital Speed:**
The orbital speed (\(v\)) of a spacecraft can be determined using the formula:
\[
v = \sqrt{\frac{GM}{r}}
\]
where:
- \(G\) is the gravitational constant (\(6.674 \times 10^{-11} \, \text{N}\,\text{m}^2\,\text{kg}^{-2}\)),
- \(M\) is the mass of the Moon (\(7.4 \times 10^{22}\) kg),
- \(r\) is the radius of the orbit, which is equal to the radius of the Moon plus the altitude of the spacecraft. Since the spacecraft is just above the lunar surface, \(r = 1700 \, \text{km} = 1.7 \times 10^6 \, \text{m}\).
2. **Orbital Period:**
The orbital period (\(T\)) can be determined using the formula:
\[
T = \frac{2\pi r}{v}
\]
After calculating the above values, the speed and period of the spacecraft in a circular orbit just above the lunar surface can be obtained.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3fe8677b-d2b4-4cf1-b1ed-08820154fcb5%2F1a793847-2d79-46a5-8d86-0575b99370ab%2F976apll_processed.png&w=3840&q=75)
Transcribed Image Text:### Orbital Mechanics of a Spacecraft around the Moon
**Problem Statement:**
The Moon's mass is \(7.4 \times 10^{22}\) kg, and its radius is 1700 km. What would be the period and the speed of a spacecraft moving in a circular orbit just above the lunar surface?
**Solution Approach:**
To solve this problem, we need to consider the fundamental principles of orbital mechanics.
1. **Orbital Speed:**
The orbital speed (\(v\)) of a spacecraft can be determined using the formula:
\[
v = \sqrt{\frac{GM}{r}}
\]
where:
- \(G\) is the gravitational constant (\(6.674 \times 10^{-11} \, \text{N}\,\text{m}^2\,\text{kg}^{-2}\)),
- \(M\) is the mass of the Moon (\(7.4 \times 10^{22}\) kg),
- \(r\) is the radius of the orbit, which is equal to the radius of the Moon plus the altitude of the spacecraft. Since the spacecraft is just above the lunar surface, \(r = 1700 \, \text{km} = 1.7 \times 10^6 \, \text{m}\).
2. **Orbital Period:**
The orbital period (\(T\)) can be determined using the formula:
\[
T = \frac{2\pi r}{v}
\]
After calculating the above values, the speed and period of the spacecraft in a circular orbit just above the lunar surface can be obtained.
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