Two planets A and B have the same uniform mass density. The radius of planet A is three times the radius of planet B . If v_A and v_B are the escape velocities on the surfaces of planets A and B respectively, then the ratio v_A v_B is :

Step-by-step solution

The escape velocity on the surface of a planet is given by v = 2GM R . Since the mass of a planet is M = 4 3 R^3 , where is the uniform mass density, we can write: v = 2G R ( 4 3 R^3 ) = R 8 G 3 For planets with the same uniform density, the escape velocity is directly proportional to the radius ( v R ). Therefore, the ratio of their escape velocities is: v_A v_B = R_A R_B Given that R_A = 3R_B , we have: v_A v_B = 3 Answer: 3