The gravitational potential energy of body of mass 'm' at the earth's surface `mgR_(e)`. Its gravitational potential energy at a height `R_(e)` fromt the earth's surface will be (here `R_(e)` is the radius of the earth)

Options

A. `-2mgR_(e)`
B. `2mgR_(e)`
C. `1/2mgR_(e)`
D. `-1/2mgR_(e)`

Explanation

To find the gravitational potential energy of a body of mass 'm' at a height equal to the radius of the Earth (Re) from the Earth's surface, we can follow these steps:

### Step-by-Step Solution:

1. **Understand Gravitational Potential Energy (U)**:
The gravitational potential energy (U) of a mass 'm' at a distance 'r' from the center of the Earth is given by the formula:
\[ U = -\frac{G M m}{r} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the Earth, and \( r \) is the distance from the center of the Earth.

2. **Calculate U at the Earth's Surface**:
At the Earth's surface, the distance from the center of the Earth is equal to the radius of the Earth (Re):
\[ U_{surface} = -\frac{G M m}{R_e} \]
This can also be expressed as:
\[ U_{surface} = -mgR_e \]
(since \( g = \frac{GM}{R_e^2} \)).

3. **Calculate U at Height Re**:
When the body is raised to a height equal to the radius of the Earth (Re), the distance from the center of the Earth becomes \( R_e + R_e = 2R_e \):
\[ U_{height} = -\frac{G M m}{2R_e} \]
This can also be expressed as:
\[ U_{height} = -\frac{mgR_e}{2} \]

4. **Find the Change in Potential Energy**:
The change in gravitational potential energy (ΔU) when moving from the surface to height Re is given by:
\[ \Delta U = U_{height} - U_{surface} \]
Substituting the values we calculated:
\[ \Delta U = \left(-\frac{mgR_e}{2}\right) - \left(-mgR_e\right) \]
Simplifying this gives:
\[ \Delta U = -\frac{mgR_e}{2} + mgR_e = \frac{mgR_e}{2} \]

5. **Final Gravitational Potential Energy at Height Re**:
Therefore, the gravitational potential energy at height Re is:
\[ U_{height} = U_{surface} + \Delta U = -mgR_e + \frac{mgR_e}{2} = -\frac{mgR_e}{2} \]

### Conclusion:
The gravitational potential energy of the body at a height equal to the radius of the Earth is:
\[ U_{height} = -\frac{1}{2} mgR_e \]