Question

Two similar-sized planets in the solar system are such that the escape velocity on one planet equals the orbital velocity of the other planet. Then the ratio of acceleration due to gravity on these planets is -
1. 4 ∶ 3
2. 1 ∶ 3
3. 2 ∶ 5
4. 1 ∶ 2

Answer 1

Correct Answer - Option 4 : 1 ∶ 2

The correct answer is option 4) i.e. 1 ∶ 2

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.

• The time period of a satellite: It is the time taken by the satellite to complete one revolution around the Earth.

Consider a satellite orbiting the earth at a height h from the surface of the earth of radius R.

The circumference of orbit of satellite = 2πR

The orbital velocity of the satellite is given by:

\(⇒ v_0 =\sqrt{\frac{GM}{R}}\)

EXPLANATION:

• Given that the planets are similarly sized i.e. they have the same radii (R).
• Let the mass of the two planets be M₁ and M₂ and the acceleration due to gravity be g₁ and g₂ respectively.
• The escape velocity = orbital velocity 

⇒ V_e = v_o

\(⇒ \sqrt{\frac{2GM_1}{R}}=\sqrt{\frac{GM_2}{R}}\)      ---(1)

• Acceleration due to gravity g is obtained from balancing the equation of force with the equation of gravitational force.

 
\(mg =\frac{GMm}{R^2}⇒ g =\frac{GM}{R^2}\)      ---(2) 

Where M is the mass of the earth, m is the mass of an object, R is the radius of the earth, and G is the gravitational constant.

Substituting (2) in (1), 

\(⇒ \sqrt{\frac{2g_1R^2}{R}}=\sqrt{\frac{g_2R^2}{R}}\)

\(\Rightarrow \frac{g_1}{g_2} = \frac{1}{2}\)