The radius and mean density of the planet are four times as that of the earth. The ratio of escape velocity at the earth to the escape velocity at a planet is

\(\begin{aligned} \mathrm{v}_{\mathrm{e}} & =\sqrt{\frac{2 \mathrm{GM}}{\mathrm{R}}} \\ \therefore \quad \mathrm{v}_{\mathrm{e}} & =\mathrm{R} \sqrt{\frac{8}{3} \pi \mathrm{G} \rho} \quad \ldots .\left(\because \mathrm{M}=\frac{4}{3} \pi \mathrm{R}^3 \rho\right)\end{aligned}\)
\(\begin{aligned} & \text { Also, } \rho_{\mathrm{p}}=4 \rho_{\mathrm{E}} \text { and } \mathrm{R}_{\mathrm{p}}=4 \mathrm{R}_{\mathrm{E}} \\ & \frac{\mathrm{v}_{\mathrm{e}}}{\mathrm{v}_{\mathrm{p}}}=\frac{\mathrm{R}_{\mathrm{E}} \sqrt{\rho_{\mathrm{E}}}}{4 \mathrm{R}_{\mathrm{E}} \sqrt{4 \rho_{\mathrm{E}}}}=\frac{1}{8}=1: 8\end{aligned}\)