Question

If two planets A and B have their densities in the ratio 2:1 and radii in the ratio 1:2, what will be the ratio of escape velocities from their surfaces?

Hint:

The escape velocity depends on the radius of the planet and its density. It is proportional to the square root of the product of density and radius squared.

Answer 1

Step 1: Use the formula for escape velocity.
The escape velocity \( v_e \) from the surface of a planet is given by the formula: \[ v_e = \sqrt{\frac{2GM}{R}} \] where:
- \( G \) is the universal gravitational constant,
- \( M \) is the mass of the planet, and
- \( R \) is the radius of the planet.

Step 2: Express mass in terms of density.
The mass of a planet is related to its density \( \rho \) and volume. The volume \( V \) of a spherical planet is given by: \[ V = \frac{4}{3} \pi R^3 \] Thus, the mass \( M \) of the planet is: \[ M = \rho \times V = \rho \times \frac{4}{3} \pi R^3 \]
Step 3: Substitute into the formula for escape velocity.
Substitute the expression for mass into the formula for escape velocity: \[ v_e = \sqrt{\frac{2G \times \rho \times \frac{4}{3} \pi R^3}{R}} = \sqrt{\frac{8 \pi G \rho R^3}{3R}} = \sqrt{\frac{8 \pi G \rho R^2}{3}} \] Thus, the escape velocity is proportional to: \[ v_e \propto \sqrt{\rho R^2} \]
Step 4: Find the ratio of escape velocities.
Now, let’s find the ratio of escape velocities for planets A and B. The ratio of escape velocities is: \[ \frac{v_{eA}}{v_{eB}} = \sqrt{\frac{\rho_A R_A^2}{\rho_B R_B^2}} \] We are given that: - \( \frac{\rho_A}{\rho_B} = \frac{2}{1} \), - \( \frac{R_A}{R_B} = \frac{1}{2} \). Substituting these values: \[ \frac{v_{eA}}{v_{eB}} = \sqrt{\frac{2 \times 1^2}{1 \times 2^2}} = \sqrt{\frac{2}{4}} = \sqrt{\frac{1}{2}} = \frac{1}{\sqrt{2}} \] Thus, the ratio of escape velocities is: \[ \boxed{\frac{1}{\sqrt{2}}} \]