Irodov Problem 1.229

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.