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: MX ∝ (2R)³ = 8R³. For Planet Y: MY ∝ R³. Therefore, vX/vY = √(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 MX = 8MY and RX = 2RY, then vX = √(2G(8MY)/(2RY)) = √(8GMY/RY) = √8 × √(GMY/RY). vY = √(2GMY/RY). So vX/vY = √8 / √2 = √4 = 2. Therefore, the ratio is 2:1.