Gravitational Potential Energy: Practice Problem & Solution

Solution Explained:

To solve this problem, we apply the core principles of Gravitational Potential Energy. Understanding the underlying formula is key to arriving at the correct answer below:

Escape velocity is given by \(v_e = R\sqrt{\frac{8}{3}\pi G\rho}\). Thus, \(v_e \propto R\sqrt{\rho}\). The ratio is \(frac{v_e}{v_p} = \frac{R_e}{R_p}\sqrt{\frac{\rho_e}{\rho_p}} = \frac{1}{2}\sqrt{\frac{1}{2}} = \frac{1}{2\sqrt{2}}\).