Two spherical planets P and Q have the same uniform density , and masses M_P and M_Q and surface areas A and 4 A respectively. Another spherical planet R also has the same uniform density , and its mass is Mp + MQ . The escape velocities from these planets is

Step-by-step solution

The surface area of a sphere is A = 4 R^2 . Given A_P = A and A_Q = 4A , we have 4 R_P^2 = A and 4 R_Q^2 = 4A . This implies R_Q^2 = 4 R_P^2 , so R_Q = 2 R_P . The mass of a planet with uniform density is M = 4 3 R^3 . Thus, M R^3 . Given R_Q = 2 R_P , we have M_Q = (2)^3 M_P = 8 M_P . The escape velocity from a planet is V = 2GM R . Since M = 4 3 R^3 , we have V = 2G 4 3 R^3 R = 8 3 G R . Thus, V R . Since R_Q = 2 R_P , it follows that V_Q = 2 V_P , so V_Q > V_P . For planet R, M_R = M_P + M_Q = M_P + 8 M_P = 9 M_