Question

Material in earth and moon is same but radius of earth is 10 times that of moon and acceleration due to gravity in earth is 6.4 times that in moon. Then the ratio of escape velocity in earth and in moon will be

• A. \( 1 : 56 \)
• B. \( 10 : 3 \)
• C. \( 6.4 : 5 \)
• D. \( 8 : 1 \)

Hint:

Escape velocity depends on the radius and mass of the celestial body. A larger radius leads to a smaller escape velocity, all else being equal.

Answer 1

The Correct Option is D

Step 1: Use the formula for escape velocity.
Escape velocity \( v_e \) is
given by the formula:
\[ v_e = \sqrt{\frac{2GM}{R}} \] where \( G \) is the gravitational constant, \( M \) is the mass of the body, and \( R \) is the radius.
Step 2: Apply the ratio.
We are given: The radius of Earth \( R_E = 10 \times R_M \) (where \( R_M \) is the radius of the Moon), The acceleration due to gravity \( g_E = 6.4 \times g_M \) (where \( g_M \) is the gravity on the Moon). Since \( g = \frac{GM}{R^2} \), the ratio of escape velocities will be: \[ \frac{v_e(Earth)}{v_e(Moon)} = \sqrt{\frac{2GM_E / R_E}{2GM_M / R_M}} = \sqrt{\frac{R_M}{R_E}} = \sqrt{\frac{1}{10}} = 8 : 1 \]
Step 3: Conclusion.
Thus, the ratio of escape velocities is \( 8 : 1 \).
Conclusion:
The correct answer is (D) \( 8 : 1 \).