CLASSICAL MECHANICS & THERMODYNAMICS - Chapter 6, Exercise 2 Solution
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The Moon's Escape Velocity

PROBLEM
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The Moon has a surface gravity of about 1.62 m/s^2 and a radius of
about 1.737 x 10^6 m. Using v = sqrt(2gr), calculate the Moon's
escape velocity, and compare it to Earth's 11.2 km/s.

SOLUTION
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Using the surface-gravity form of the escape velocity formula:

    v = sqrt(2gr)

    g = 1.62 m/s^2
    r = 1.737 x 10^6 m

    v = sqrt(2 x 1.62 x 1.737 x 10^6)
    v = sqrt(5.628 x 10^6)
    v = 2372 m/s (approximately)
    v = 2.37 km/s (approximately)

ANSWER: The Moon's escape velocity is approximately 2.37 km/s -
roughly one-fifth (about 21%) of Earth's 11.2 km/s.

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WHY THIS WORKS AS AN ANSWER

Escape velocity depends on both the strength of surface gravity (g)
and the size (radius, r) of the body - a body needs a combination of
strong enough gravity AND enough "distance to climb out of" for
escape to be hard. The Moon has both a much weaker surface gravity
than Earth (about 1/6th) and a smaller radius (roughly 27% of
Earth's), and both factors combine inside the square root to make the
Moon's escape velocity far lower than Earth's. This is exactly why
the real Apollo lunar module needed only a comparatively small ascent
engine to leave the Moon's surface and return to lunar orbit, while a
rocket leaving Earth needs vastly more powerful engines to reach
11.2 km/s.