Question

If the ratio of escape velocities is 3:2 from two different planets A and B of radii in the ratio 2:3, find the ratio of acceleration due to gravity at the surface of A to that at the surface of B.

Hint:

The escape velocity is directly related to the acceleration due to gravity and the radius of the planet. Use the ratio of escape velocities and radii to find the ratio of accelerations.

Answer 1

Step 1: Formula for escape velocity.
The escape velocity \( v_e \) from 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.
Step 2: Relating escape velocities and radii.
We are given that the ratio of escape velocities from planets A and B is \( 3:2 \), and the ratio of their radii is \( 2:3 \). Let \( v_{eA} \) and \( v_{eB} \) be the escape velocities, and \( R_A \) and \( R_B \) be the radii of planets A and B, respectively. We know: \[ \frac{v_{eA}}{v_{eB}} = \frac{3}{2}, \quad \frac{R_A}{R_B} = \frac{2}{3} \]
Step 3: Express escape velocity in terms of acceleration due to gravity.
The escape velocity is related to the acceleration due to gravity \( g \) at the surface of a planet by the formula: \[ v_e = \sqrt{2gR} \] Therefore, for planets A and B: \[ v_{eA} = \sqrt{2g_A R_A}, \quad v_{eB} = \sqrt{2g_B R_B} \] Squaring both equations: \[ v_{eA}^2 = 2g_A R_A, \quad v_{eB}^2 = 2g_B R_B \] Taking the ratio of the escape velocities squared: \[ \frac{v_{eA}^2}{v_{eB}^2} = \frac{g_A R_A}{g_B R_B} \] Using the given ratios: \[ \left( \frac{3}{2} \right)^2 = \frac{g_A R_A}{g_B R_B} \] Simplifying: \[ \frac{9}{4} = \frac{g_A R_A}{g_B R_B} \] Substitute \( \frac{R_A}{R_B} = \frac{2}{3} \): \[ \frac{9}{4} = \frac{g_A \cdot \frac{2}{3}}{g_B} \] Solving for \( \frac{g_A}{g_B} \): \[ \frac{g_A}{g_B} = \frac{9}{4} \cdot \frac{3}{2} = \frac{27}{8} \]
Step 4: Final ratio.
Thus, the ratio of the acceleration due to gravity at the surface of A to that at the surface of B is: \[ \frac{g_A}{g_B} = \frac{27}{8} \]