Question

What is the ratio of escape velocity to orbital velocity?

• A. \(1:1\)
• B. \(2:1\)
• C. \(\sqrt{2}:1\)
• D. \(1:\sqrt{2}\)

Hint:

Escape velocity is \(\sqrt{2}\) times the orbital velocity for the same celestial body.

Answer 1

The Correct Option is C

Concept: Escape velocity is the minimum velocity required for an object to escape the gravitational field of a planet without returning. \[ v_e = \sqrt{\frac{2GM}{R}} \] Orbital velocity is the velocity required for a satellite to remain in circular orbit around a planet. \[ v_o = \sqrt{\frac{GM}{R}} \] where \(G\) = gravitational constant, \(M\) = mass of the planet, \(R\) = radius of the planet.

Step 1: Write the ratio of escape velocity to orbital velocity. \[ \frac{v_e}{v_o} = \frac{\sqrt{\frac{2GM}{R}}}{\sqrt{\frac{GM}{R}}} \]

Step 2: Simplify the expression. \[ \frac{v_e}{v_o} = \sqrt{2} \] Thus the ratio is \[ \sqrt{2} : 1 \]