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Two spherical planets P and Q have the same uniform density rho, masses M_P and M_Q, and surface areas A and 4A respectively. A third spherical planet R also has uniform density rho and its mass equals M_P + M_Q. The escape velocities from P, Q, and R are V_P, V_Q, and V_R respectively. Which of the following is/are correct?
- V_Q > V_R > V_P
- V_R > V_Q > V_P
- V_R / V_P = 3
- V_P / V_Q = 1/2
Correct answer: V_R > V_Q > V_P
Solution
Since V_esc proportional to R for uniform density, and R_Q = 2R_P, R_R = 9^(1/3) * R_P approximately 2.08 R_P, we get V_R > V_Q > V_P. Also V_P/V_Q = R_P/(2R_P) = 1/2 is also correct, making both (b) and (d) valid.
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