If the radius of a planet is \(\alpha\) times the radius of the earth and its mass \(\beta\) times that of the earth, find the ratio of the acceleration of the force of gravity and the first and second cosmic velocities to the corresponding quantities for the earth.

To find the gravitational acceleration on the planet, we will use the formula: \(g = \frac{GM}{R^2}\) where G is the gravitational constant, M is the mass of the planet, and R is its radius. Since the mass of the planet is β times the Earth's mass (Me) and its radius is α times the Earth's radius (Re), we have: \(g_p = \frac{G\cdot(βM_e)}{(αR_e)^2}\)

The first cosmic velocity (orbital velocity) for the planet can be found using the following formula: \(v_{1_p} = \sqrt{\frac{GM}{R}}\) Substituting the values for the planet's mass and radius, we get: \(v_{1_p} = \sqrt{\frac{G\cdot(βM_e)}{αR_e}}\)

The second cosmic velocity (escape velocity) for the planet is given by the formula: \(v_{2_p} = \sqrt{\frac{2GM}{R}}\) Substituting the values for the planet's mass and radius, we get: \(v_{2_p} = \sqrt{\frac{2G\cdot(βM_e)}{αR_e}}\)

Now, we need to find the ratio of the gravitational acceleration, first cosmic velocity, and second cosmic velocity for the planet to the corresponding quantities for Earth. The gravitational acceleration on Earth is: \(g_e = \frac{GM_e}{R_e^2}\) Thus, the ratio of gravitational acceleration for the planet and Earth is: \(\frac{g_p}{g_e} = \frac{\frac{G\cdot(βM_e)}{(αR_e)^2}}{\frac{GM_e}{R_e^2}} = \frac{βM_e}{α^2R_e^2}\cdot\frac{R_e^2}{M_e} = \frac{β}{α^2}\) The first cosmic velocity on Earth is: \(v_{1_e} = \sqrt{\frac{GM_e}{R_e}}\) Thus, the ratio of the first cosmic velocity for the planet and Earth is: \(\frac{v_{1_p}}{v_{1_e}} = \frac{\sqrt{\frac{G\cdot(βM_e)}{αR_e}}}{\sqrt{\frac{GM_e}{R_e}}} = \sqrt{\frac{βM_e}{αR_e}\cdot\frac{1}{M_e}\cdot\frac{R_e}{1}} = \sqrt{\frac{β}{α}}\) The second cosmic velocity on Earth is: \(v_{2_e} = \sqrt{\frac{2GM_e}{R_e}}\) Thus, the ratio of the second cosmic velocity for the planet and Earth is: \(\frac{v_{2_p}}{v_{2_e}} = \frac{\sqrt{\frac{2G\cdot(βM_e)}{αR_e}}}{\sqrt{\frac{2GM_e}{R_e}}} = \sqrt{\frac{2\cdotβM_e}{αR_e}\cdot\frac{1}{M_e}\cdot\frac{R_e}{2}} = \sqrt{\frac{β}{α}}\) So the ratios of the acceleration due to gravity and the first and second cosmic velocities for the planet to the corresponding quantities for Earth are: Gravitational acceleration ratio: \(\frac{β}{α^2}\) First cosmic velocity ratio: \(\sqrt{\frac{β}{α}}\) Second cosmic velocity ratio: \(\sqrt{\frac{β}{α}}\)