Two planets, Planet X and Planet Y, have the same density but Planet X has twice the radius of Planet Y. What is the ratio of the escape velocity from Planet X to the escape velocity from Planet Y?

Core Concept

Escape velocity v = √(2GM/R). Since density ρ = M/(4/3)πR³ is constant, M ∝ R³. For Planet X: M_X ∝ (2R)³ = 8R³. For Planet Y: M_Y ∝ R³. Therefore, v_X/v_Y = √(2G(8R³)/(2R)) / √(2GR³/R) = √(8R²/R) / √(R²/R) = √(8R) / √R = √8 = 2√2. However, this doesn't match any options. Let me reconsider: v = √(2GM/R). If M_X = 8M_Y and R_X = 2R_Y, then v_X = √(2G(8M_Y)/(2R_Y)) = √(8GM_Y/R_Y) = √8 × √(GM_Y/R_Y). v_Y = √(2GM_Y/R_Y). So v_X/v_Y = √8 / √2 = √4 = 2. Therefore, the ratio is 2:1.