Physics - Gravitation Question with Solution | TestHub

The escape velocity is $V_e=\sqrt{\frac{2GM}{R}}$.

Since mass $M=\rho \cdot \frac{4}{3}\pi R^3$, we have $V_e=\sqrt{\frac{2G(\rho \frac{4}{3}\pi R^3)}{R}}=\sqrt{\frac{8G\pi \rho R^2}{3}}$.

Thus, $V_e\propto R\sqrt{\rho }$.

If radius is $2R$ and density is same, $V_e^{\prime }\propto (2R)\sqrt{\rho }=2(R\sqrt{\rho })=2V_e$.

So, $X=2$.