Two spherical planets, P and Q , have different radii and uniform densities. The escape velocity from the surface of planet P is half the escape velocity from the surface of planet Q . If the radius of planet P is twice the radius of planet Q , the ratio of the density of planet P to the density of planet Q is:

Step-by-step solution

The escape velocity v_e from the surface of a spherical planet of radius R and uniform density is given by: v_e = 2GM R = 2G R ( 4 3 R^3 ) = R 8 G 3 Thus, v_e R . Rearranging for density, we get v_e^2 R^2 . For planets P and Q , the ratio of their densities is: _P _Q = ( v_P v_Q )^2 ( R_Q R_P )^2 Given that v_P = 1 2 v_Q and R_P = 2R_Q , we substitute these values: _P _Q = ( 1 2 )^2 ( 1 2 )^2 = 1 4 1 4 = 1 16 The ratio of their densities is 1 : 16 . Answer: 1 : 16