Planet A has a radius R , uniform density d , and an escape velocity v_e from its surface. Planet B has a radius 2R and the same uniform density d . If a body is projected from the surface of Planet B with a velocity of 3v_e , what will be the velocity of this body at an infinite distance from Planet B?

Step-by-step solution

The escape velocity from the surface of a planet of radius R and uniform density d is given by: v_e = 2GM R = 2G R ( 4 3 R^3 d ) = R 8 3 G d Since density is constant, the escape velocity is directly proportional to the radius of the planet ( v_e R ). For Planet B, the radius is 2R , so its escape velocity is v_ eB = 2v_e . By the principle of conservation of mechanical energy for the body projected from Planet B: 1 2 m v_ proj ^2 - GM_B m R_B = 1 2 m v_ ^2 + 0 v_ ^2 = v_ proj ^2 - v_ eB ^2 Substitute v_ proj = 3v_