ASTRONOMY FUNDAMENTALS - Chapter 3, Exercise 2 Solution
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How Mass and Radius Each Affect Escape Velocity

PROBLEM
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Using v = sqrt(2GM/r), explain what happens to escape velocity if a
planet's mass doubles while its radius stays the same, versus what
happens if its radius doubles while its mass stays the same. Which
change increases escape velocity, and which decreases it?

SOLUTION
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Case 1: Mass doubles, radius stays the same (M -> 2M, r unchanged)

    v_new = sqrt(2G(2M)/r) = sqrt(2 x (2GM/r)) = sqrt(2) x sqrt(2GM/r)
    v_new = sqrt(2) x v_original
    v_new ≈ 1.41 x v_original

    Escape velocity INCREASES, by a factor of the square root of 2
    (about 41% higher).

Case 2: Radius doubles, mass stays the same (M unchanged, r -> 2r)

    v_new = sqrt(2GM/(2r)) = sqrt((1/2) x (2GM/r)) = sqrt(1/2) x sqrt(2GM/r)
    v_new = v_original / sqrt(2)
    v_new ≈ 0.71 x v_original

    Escape velocity DECREASES, to about 71% of its original value.

ANSWER: Doubling the mass increases escape velocity (by a factor of
sqrt(2)); doubling the radius decreases escape velocity (by a factor of
1/sqrt(2)).

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WHY THIS WORKS AS AN ANSWER

Mass sits in the numerator of the formula, so increasing it directly
increases escape velocity - a more massive body has stronger gravity to
escape from, requiring more speed. Radius sits in the denominator, so
increasing it decreases escape velocity - a larger radius means the
surface is farther from the body's own center of mass, where gravity is
weaker (gravity falls off with the square of distance), so less speed
is needed to escape from that more distant starting point.

Because the formula takes a square root of the whole expression, mass
and radius don't affect escape velocity by the same simple ratio they
themselves changed by - doubling either one changes escape velocity by
a factor of sqrt(2), not a factor of 2, since the square root
"compresses" the effect of whatever is inside it.