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:
We are given ratios of escape velocities and radii for two planets. We need to find the ratio of their surface gravities by connecting these three quantities algebraically.
Step 2: Key Formula or Approach:
The escape velocity formula in terms of surface gravity is \(v_e = \sqrt{2gR}\).
Rearranging this gives: \(g = \frac{v_e^2}{2R}\).
Use this to form a ratio: \(\frac{g_A}{g_B} = \left(\frac{v_{eA}}{v_{eB}}\right)^2 \times \frac{R_B}{R_A}\).
Step 3: Detailed Explanation:
Given ratios:
\(\frac{v_{eA}}{v_{eB}} = \frac{2}{3}\)
\(\frac{R_A}{R_B} = \frac{3}{4} \implies \frac{R_B}{R_A} = \frac{4}{3}\)
Substitute these directly into the gravity ratio equation:
\[ \frac{g_A}{g_B} = \left(\frac{2}{3}\right)^2 \times \left(\frac{4}{3}\right) \] \[ \frac{g_A}{g_B} = \frac{4}{9} \times \frac{4}{3} \] \[ \frac{g_A}{g_B} = \frac{16}{27} \] Step 4: Final Answer:
The ratio is 16 : 27.