➡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.
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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