Question

A spherically uniform planet of mass \( 8 \times 10^{24} \) kg and of radius \( 6 \times 10^6 \) m is orbiting around the Sun. The escape velocity for the planet is close to (Take \( G = 6 \times 10^{-11} \text{ N m}^2\text{kg}^{-2} \)) (A) 11.2 km/s

• A. 16 km/s
• B. 4 km/s
• C. 12.6 km/s
• D. 1.6 km/s

Hint:

Escape velocity depends only on \(M\) and \(R\).

Answer 1

The Correct Option is D

Concept: Escape velocity
Escape velocity is the minimum speed required for an object to escape a planet’s gravitational field without further propulsion: \[ v = \sqrt{\frac{2GM}{R}} \]

Step 1: Substitute known values
For Earth: \[ G = 6.67 \times 10^{-11}, \quad M = 6 \times 10^{24} \text{ kg}, \quad R = 6.4 \times 10^6 \text{ m} \]

Step 2: Calculate
\[ v = \sqrt{\frac{2 \times 6.67 \times 10^{-11} \times 6 \times 10^{24}}{6.4 \times 10^6}} \] \[ v \approx 1.26 \times 10^4 \text{ m/s} \]

Step 3: Convert units
\[ v \approx 12.6 \text{ km/s} \] Final Answer:
\[ \boxed{12.6 \text{ km/s}} \] Note:
Escape velocity is independent of the mass of the object being projected.