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A rocket must be launched from Earth so that it never returns (escapes). The minimum energy the launcher must supply for this from Earth is E. If the same rocket is instead launched to escape from the Moon's surface, what minimum energy is required? Assume Earth and Moon have equal densities and that Earth's volume is 64 times the Moon's volume.
- E/4
- E/16
- E/32
- E/64
Correct answer: E/16
Solution
Escape energy E = (1/2) m Vₑ² = (1/2) m (2GM/R) = GMm/R. For equal density, M = (4/3)pi R³ rho, so E ~ M/R ~ R³/R = R². Volume ratio Earth:Moon = 64 => radius ratio = 64^(1/3) = 4, so R_moon = R_earth/4. Therefore E_moon/E_earth = (R_moon/R_earth)² = (1/4)² = 1/16. Hence E_moon = E/16.
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