Question

The escape velocity 'v' of a planet is related to its mean density 'ρ' as -
1. v ∝ ρ 
2. v ∝ √ρ 
3. v ∝ ρ^-2 
4. v ∝ ρ³ 

Answer 1

Correct Answer - Option 2 : v ∝ √ρ 

The correct answer is option 1) i.e. v ∝ √ρ 

CONCEPT:

• Escape velocity is the minimum velocity with which a body is projected from the surface of the planet so as to reach infinity, by overcoming the pull by gravity.

Escape velocity at the surface of a planet is given by:

 \(⇒ V_e=\sqrt{\frac{2GM}{R}}\)  

Where G = gravitational constant (6.67 × 10-11 Nm2/kg2), M = mass of the planet and R = radius of the planet.

EXPLANATION:

We know that mass (M) = density (ρ) × volume (V)

Mass of a planet, \(M = \rho \times \frac{4}{3}\pi R^3\)

Escape speed, \(v=\sqrt{\frac{2GM}{R}} = \sqrt{\frac{2G(\rho \times \frac{4}{3}\pi R^3)}{R}}\)

⇒ V ∝ √ρ