Question

The escape velocities of two planets \(A\) and \(B\) are in the ratio \(1: 2\) If the ratio of their radii respectively is\(1: 3\), then the ratio of acceleration due to gravity of planet \(A\) to the acceleration of gravity of planet B will be :

• A. \(\frac{3}{2}\)
• B. \(\frac{2}{3}\)
• C. \(\frac{3}{4}\)
• D. \(\frac{4}{3}\)

Hint:

Remember the formulas for escape velocity and acceleration due to gravity. Expressing the mass of a planet in terms of its density and radius can be helpful in solving such problems.

Answer 1

The Correct Option is C

Step 1: Recall the Formula for Escape Velocity

The escape velocity (\(V_e\)) of a planet is given by:

\[ V_e = \sqrt{\frac{2GM}{R}} \]

where \(G\) is the gravitational constant, \(M\) is the mass of the planet, and \(R\) is the radius of the planet. We can also express the mass \(M\) in terms of density (\(\rho\)) and volume:

\[ M = \rho \times \frac{4}{3}\pi R^3 \]

Substituting this into the escape velocity formula gives:

\[ V_e = \sqrt{\frac{2G(\rho \times \frac{4}{3}\pi R^3)}{R}} = \sqrt{\frac{8G\pi}{3} R^2} = C\sqrt{\rho R} \]

where \(C\) is a constant.

Step 2: Set Up the Ratio of Escape Velocities

Let \(V_{e1}\) and \(V_{e2}\) be the escape velocities of planets A and B respectively, and let \(R_1\) and \(R_2\) be their radii, and \(\rho_1\) and \(\rho_2\) their densities. Given:

\[ \frac{V_{e1}}{V_{e2}} = \frac{C\sqrt{\rho_1R_1}}{C\sqrt{\rho_2R_2}} = \frac{1}{2} \]

and

\[ \frac{R_1}{R_2} = \frac{1}{3}, \]

we have:

\[ \sqrt{\frac{\rho_1R_1}{\rho_2R_2}} = \frac{1}{2} \] \[ \frac{\rho_1R_1}{\rho_2R_2} = \frac{1}{4} \] \[ \frac{\rho_1}{\rho_2} = \frac{1}{4} \times \frac{R_2}{R_1} = \frac{1}{4} \times 3 = \frac{3}{4}. \]

Step 3: Recall the Formula for Acceleration Due to Gravity

The acceleration due to gravity (\(g\)) on a planet is given by:

\[ g = \frac{GM}{R^2} = \frac{G \times \frac{4}{3}\pi R^3 \rho}{R^2} = \frac{4\pi G \rho R}{3} = C\rho R, \]

where \(C = \frac{4\pi G}{3}\) is a constant.

Step 4: Find the Ratio of Accelerations Due to Gravity

Let \(g_1\) and \(g_2\) be the accelerations due to gravity on planets A and B, respectively. Then:

\[ \frac{g_1}{g_2} = \frac{C\rho_1R_1}{C\rho_2R_2} = \frac{\rho_1R_1}{\rho_2R_2} = \frac{1}{4} \times \frac{R_2}{R_1} = \frac{R_1}{R_2} \times \frac{\rho_1}{\rho_2} = \frac{1}{4} \times 3 = \frac{3}{4}. \]

Conclusion:

The ratio of the acceleration due to gravity of planet A to that of planet B is \(\frac{3}{4}\) (Option 3).