Question

The escape velocities of two planets $A$ and $B$ are in the ratio $2 : 3$. If the ratio of their radii is $3 : 4$, then the ratio of acceleration due to gravity at the surface of the planet $A$ to that at the surface of the planet $B$ is

• A. $4 : 9$
• B. $4 : 3$
• C. $16 : 27$
• D. $4 : 27$
• E. $16 : 9$

Hint:

When dealing with ratios in gravitation, writing the formula in terms of proportionalities (\( g \propto v^2/R \)) prevents confusion between numerator and denominator.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Escape velocity depends on the gravity at the surface and the radius of the planet. We can relate these physical quantities to find the desired ratio.
Key Formula or Approach:
The escape velocity \( v_e \) is given by:
\[ v_e = \sqrt{2gR} \implies v_e^2 = 2gR \]
So, gravity \( g \propto \frac{v_e^2}{R} \).

Step 2: Detailed Explanation:
Let the ratios be:
\( \frac{v_A}{v_B} = \frac{2}{3} \)
\( \frac{R_A}{R_B} = \frac{3}{4} \)
The ratio of gravity is:
\[ \frac{g_A}{g_B} = \left( \frac{v_A}{v_B} \right)^2 \times \frac{R_B}{R_A} \]
Substitute the given values:
\[ \frac{g_A}{g_B} = \left( \frac{2}{3} \right)^2 \times \frac{4}{3} \]
\[ \frac{g_A}{g_B} = \frac{4}{9} \times \frac{4}{3} = \frac{16}{27} \]

Step 3: Final Answer:
The ratio of acceleration due to gravity is $16 : 27$.