Problem Statement
A spacecraft approaches the Moon (mass M, radius R) from far away. Find the minimum speed needed to orbit just above the Moon’s surface without rocket propulsion.
Given
M (Moon), R. First cosmic velocity for Moon.
Concepts & Formulas
Orbital speed at surface: v = √(GM/R). No atmosphere → orbit is possible at surface level.
Step-by-Step Solution
Step 1: Centripetal: mv²/R = GMm/R² → v = √(GM/R).
Step 2: Using g_Moon = GM/R²: v = √(g_Moon·R).
Step 3: Moon: g_Moon ≈ 1.62 m/s², R ≈ 1.74×10⁶ m → v ≈ 1.68 km/s.
Worked Calculation
v = √(GM/R) = √(g_Moon·R) ≈ 1.68 km/s.
Boxed Answer
v = √(GM/R) ≈ 1.68 km/s
Physical Interpretation
The Moon’s first cosmic speed is about 1/5 of Earth’s because its lower gravity and smaller radius both reduce the required speed.
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