Gravitational Potential Energy

Step 1: Understanding the given condition
The gravitational potential energy at a height is given by: where: - is the universal gravitational constant,
- is the mass of the Earth,
- is the mass of the body,
- is the radius of the Earth,
- is the height above the Earth's surface.
Step 2: Expressing in terms of given height
Given that , the new gravitational potential energy is: Step 3: Expressing in terms of initial potential energy
The gravitational potential energy on the Earth's surface is: Thus, Thus, the correct answer is option (A) .

Step 1: Understanding the Energy of a Satellite in Orbit - The total mechanical energy of a satellite in a circular orbit is given by: - The kinetic energy of the satellite is given by: - The potential energy of the satellite in orbit is: - The total energy of the system is: - Since we are given that kinetic energy is , we write: Thus, the total energy of the satellite is:
Step 2: Energy Required for Escape - For the satellite to escape Earth's gravitational field, its total energy must be zero (i.e., it must reach infinity with zero velocity).
- The energy required to remove the satellite from orbit is: Substituting , Thus, the minimum energy required to make the satellite escape from the Earth is: