On planet A the escape speed of a body is 12 km/s. Planet B has four times the density of A and half the radius of A. What is the escape speed of the body on planet B?

12 km/s
24 km/s
6 km/s
36 km/s

Correct answer: 12 km/s

Solution

Escape velocity v = sqrt(2*G*M/R) and M = (4/3)*pi*R³*rho, so v = R*sqrt(8*pi*G*rho/3), i.e. v ~ R*sqrt(rho). For B: R_B = R_A/2, rho_B = 4*rho_A, so v_B/v_A = (R_B/R_A)*sqrt(rho_B/rho_A) = (1/2)*sqrt(4) = (1/2)*2 = 1. Thus v_B = v_A = 12 km/s.

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