Two planets have escape velocities in the ratio 2 : 1 and acceleration due to gravity at their surfaces in the ratio 1 : 2 . The ratio of their radii is:

Step-by-step solution

The escape velocity v_e is related to the acceleration due to gravity g and radius R by the formula v_e = 2gR . Rearranging this for the radius gives R = v_e^2 2g . Therefore, the ratio of the radii of the two planets is: R₁ R₂ = ( v₁ v₂ )^2 ( g₂ g₁ ) Given v₁ v₂ = 2 1 and g₁ g₂ = 1 2 (which implies g₂ g₁ = 2 1 ): R₁ R₂ = (2)^2 (2) = 4 2 = 8 The ratio of their radii is 8 : 1 . Answer: 8 : 1