Question

A planet 'A' having density \( \rho \) and radius \( R \) has escape velocity \( = 10 \, \text{km/sec} \). Find the escape velocity of a planet B having density and radius both 10% that of planet A.

• A. \( \frac{1}{\sqrt{10}} \)
• B. \( \frac{1}{\sqrt{20}} \)
• C. \( \frac{1}{\sqrt{30}} \)
• D. \( \frac{1}{\sqrt{50}} \)

Hint:

The escape velocity depends on the square root of the mass-to-radius ratio, so changes in radius and density affect the velocity significantly.

Answer 1

The Correct Option is A

Step 1: Formula for escape velocity.
The escape velocity \( v_e \) is given by: \[ v_e = \sqrt{\frac{2GM}{R}} \] where \( G \) is the gravitational constant, \( M \) is the mass, and \( R \) is the radius of the planet. Step 2: Apply the given conditions.
The mass of a planet is given by \( M = \rho R^3 \). Hence, the escape velocity can be written as: \[ v_e = \sqrt{\frac{2G \rho R^3}{R}} = \sqrt{2G \rho R^2} \] For planet B, both the radius and density are 10% that of planet A. Therefore, the escape velocity for planet B is: \[ v_{eB} = \sqrt{2G (0.1 \rho) (0.1 R)^2} = \frac{1}{\sqrt{10}} v_{eA} \] Step 3: Conclusion.
The escape velocity of planet B is \( \frac{1}{\sqrt{10}} \) of the escape velocity of planet A. Final Answer: \[ \boxed{\frac{1}{\sqrt{10}}} \]