Question

The gravitational potential energy of a body on the surface of the Earth is \(E\). If the body is taken from the surface of the Earth to a height equal to \(150%\) of the radius of the Earth, its gravitational potential energy is:

• A. \(0.4E \)
• B. \(0.2E \)
• C. \(0.6E \)
• D. \(0.3E \)

Hint:

The gravitational potential energy at height \( h \) is given by: \[ U = \frac{U_0}{1 + \frac{h}{R}} \] where \( U_0 \) is the gravitational potential energy on the Earth's surface.

Answer 1

The Correct Option is A

Step 1: Understanding the given condition
The gravitational potential energy at a height \(h\) is given by: \[ U = - \frac{GMm}{R + h} \] where: - \( G \) is the universal gravitational constant,
- \( M \) is the mass of the Earth,
- \( m \) is the mass of the body,
- \( R \) is the radius of the Earth,
- \( h \) is the height above the Earth's surface.
Step 2: Expressing in terms of given height
Given that \( h = 1.5 R \), the new gravitational potential energy is: \[ U' = - \frac{GMm}{R + 1.5R} = - \frac{GMm}{2.5R} \] Step 3: Expressing in terms of initial potential energy
The gravitational potential energy on the Earth's surface is: \[ U_0 = - \frac{GMm}{R} \] Thus, \[ U' = \frac{U_0}{2.5} = 0.4 U_0 \] Thus, the correct answer is option (A) \(0.4E\).