Physics - Gravitation Question with Solution | TestHub

According to the question $$R_P=2R_E,{\rho }_E={\rho }_P$$ Escape velocity of earth $$\begin{array}{rl}{V}_E& =\sqrt{\frac{2G{M}_E}{R_E}}=\sqrt{\frac{2G}{R_E}\left(\frac{4}{3}\pi R_E^3{\rho }_E\right)}\\ & =R_E\sqrt{\frac{8}{3}\pi G{\rho }_E}\end{array}$$ Similarly Escape velocity of planet $$\begin{array}{rl}{V}_P& =\sqrt{\frac{2G{M}_P}{R_P}}=\sqrt{\frac{2G}{R_P}\left(\frac{4}{3}\pi R_p{}^3{\rho }_p\right)}\\ & =R_P\sqrt{\frac{8}{3}\pi G{\rho }_p}\end{array}$$ from eq. (i) and eq. (ii) $$\frac{V_E}{V_P}=\frac{R_E}{R_P}\sqrt{\frac{{\rho }_E}{{\rho }_P}}=\frac{R_E}{2R_E}\sqrt{\frac{{\rho }_Q}{{\rho }_E}}=\frac{1}{2}$$ (given) $$\Rightarrow V_p=2V_E$$