A planet A has an escape velocity v₀ and acceleration due to gravity at its surface g₀ . Another planet B has a uniform mass density twice that of planet A and an acceleration due to gravity at its surface four times that of planet A . The escape velocity of planet B is

Step-by-step solution

The acceleration due to gravity at the surface of a planet is given by g = GM R^2 . Writing mass in terms of uniform density , we get M = 4 3 R^3 . Thus, g = G R^2 ( 4 3 R^3 ) = 4 3 G R This implies that the radius R is proportional to g . The escape velocity is given by v_e = 2gR . Substituting the proportionality for R , we get: v_e g ( g ) = g For planet B , we are given g_B = 4g_A and _B = 2 _A . Therefore, the escape velocity of planet B is: v_B = v_A ( g_B g_A ) _A _B = v₀ (4) 1 2 = 2 2 v₀ Answer: 2 2 v₀