91Ó°ÊÓ

The formula for density is \[\rho = \frac{m}{V}\] Since both planets have the same mass and their densities have a ratio of 1:8, we can use this ratio to find the relationship between their volumes \(V_1\) and \(V_2\). Let: Density of Planet 1: \(\rho_1\) Density of Planet 2: \(\rho_2\) Volume of Planet 1: \(V_1\) Volume of Planet 2: \(V_2\) Mass of Planets: \(m\) Given that \(\frac{\rho_1}{\rho_2} = \frac{1}{8}\), using the density formula we get the corresponding ratio of their volumes: \[\frac{m/V_1}{m/V_2} = \frac{1}{8}\] \[\frac{V_2}{V_1} = \frac{1}{8}\] Now, we can use the formula for the volume of a sphere to relate their radii \(r_1\) and \(r_2\): \[V = \frac{4}{3}\pi r^3\] \[\frac{4}{3}\pi r_2^3 = \frac{1}{8} \cdot \frac{4}{3}\pi r_1^3\] Canceling common terms, we find the ratio of their radii: \[\frac{r_2^3}{r_1^3} = \frac{1}{8}\] Taking the cube root of both sides: \[\frac{r_2}{r_1} = \frac{1}{2}\]

To calculate the ratio of gravitational acceleration of the two planets, we use the gravitational acceleration formula: \[g = \frac{Gm}{r^2}\] Taking the ratio of g for both planets, we get: \[\frac{g_1}{g_2} = \frac{Gm / r_1^2}{Gm / r_2^2}\] Canceling common terms: \[\frac{g_1}{g_2} = \frac{r_2^2}{r_1^2}\] Using the radius ratio we found earlier: \[\frac{g_1}{g_2}= \frac{\left(\frac{1}{2}r_1\right)^2}{r_1^2}= \frac{1}{4}\] Therefore, the acceleration due to gravity ratio is 1:4, which corresponds to option (B).

To determine the escape velocity ratio, we can use the escape velocity formula: \[v_e = \sqrt{\frac{2Gm}{r}}\] Taking the ratio of escape velocities for the two planets, we get: \[\frac{v_{e1}}{v_{e2}} = \frac{\sqrt{2Gm / r_1}}{\sqrt{2Gm / r_2}}\] Canceling common terms: \[\frac{v_{e1}}{v_{e2}} = \sqrt{\frac{r_2}{r_1}}\] Using the radius ratio we found earlier: \[\frac{v_{e1}}{v_{e2}} = \sqrt{\frac{1/2r_1}{r_1}} = \sqrt{1/2} = \sqrt{2}:1\] Thus, the escape velocity ratio is \(\sqrt{2}:1\), which corresponds to option (C). In conclusion, the correct answer is that the acceleration due to gravity ratio is 1:4 (B), and the escape velocity ratio is \(\sqrt{2}:1\) (C).