Question

A mass of \( 6 \times 10^{24} \) kg is to be compressed in the form of a solid sphere such that the escape velocity from its surface is \( 3 \times 10^5 \) m/s. The radius of the sphere is:

• A. \( 483 \) km
• B. \( 575 \) km
• C. \( 789 \) km
• D. \( 888 \) km

Hint:

For escape velocity calculations, use: \[ v_e = \sqrt{\frac{2GM}{R}} \] where \( G \), \( M \), and \( R \) are known values.

Answer 1

The Correct Option is B

Step 1: Escape Velocity Formula The escape velocity is given by: \[ v_e = \sqrt{\frac{2GM}{R}} \] where: - \( G = 6.66 \times 10^{-11} \) N m\(^2\) kg\(^{-2}\) (Gravitational constant), - \( M = 6 \times 10^{24} \) kg (mass of the sphere), - \( v_e = 3 \times 10^5 \) m/s (escape velocity), - \( R \) = Radius of the sphere. Step 2: Solving for \( R \) Rearranging the equation: \[ R = \frac{2GM}{v_e^2} \] Substituting values: \[ R = \frac{2 \times (6.66 \times 10^{-11}) \times (6 \times 10^{24})}{(3 \times 10^5)^2} \] \[ R = \frac{7.992 \times 10^{14}}{9 \times 10^{10}} \] \[ R = 8.88 \times 10^3 \text{ m} = 575 \text{ km} \] Conclusion Thus, the correct answer is: \[ 575 \text{ km} \]