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

Step 1: Understanding the Question:
The question asks for the mathematical relationship/ratio between the velocity needed to leave a planet's gravity (\(v_e\)) and the velocity needed to orbit it (\(v_o\)).
Step 2: Key Formula or Approach:
The formulas for these velocities for a mass at the surface (or near surface) are:
\[ v_e = \sqrt{\frac{2GM}{R}} \]
\[ v_o = \sqrt{\frac{GM}{R}} \]
Step 3: Detailed Explanation:
To find the ratio, we divide the expression for escape velocity by the expression for orbital velocity:
\[ \text{Ratio} = \frac{v_e}{v_o} = \frac{\sqrt{\frac{2GM}{R}}}{\sqrt{\frac{GM}{R}}} \]
Extracting the constants from the square root:
\[ \frac{v_e}{v_o} = \frac{\sqrt{2} \cdot \sqrt{\frac{GM}{R}}}{\sqrt{\frac{GM}{R}}} \]
Canceling out the common terms:
\[ \frac{v_e}{v_o} = \sqrt{2} \]
Step 4: Final Answer:
The ratio is \(\sqrt{2} : 1\).