The gravitational potential energy of a body of mass ‘ m ’ at the earth’s surface `-mgR_(e)`. Its gravitational potential energy at a height `R_(e)` from the earth’s surface will be (Here `R_(e)` is the radius of the earth)

Options

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

Answer

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 at the Surface:**
The gravitational potential energy (U) of a body of mass 'm' at the Earth's surface is given by:
\[ U = -\frac{G M m}{R_e} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the Earth, and \( R_e \) is the radius of the Earth.

2. **Determine the Height Above the Surface:**
We need to find the gravitational potential energy at a height \( R_e \) above the Earth's surface. At this height, the distance from the center of the Earth becomes:
\[ d = R_e + R_e = 2R_e \]

3. **Use the Formula for Gravitational Potential Energy:**
The gravitational potential energy at a distance \( d \) from the center of the Earth is given by:
\[ U = -\frac{G M m}{d} \]
Substituting \( d = 2R_e \):
\[ U = -\frac{G M m}{2R_e} \]

4. **Relate \( G \) and \( g \):**
We know that the acceleration due to gravity at the surface of the Earth is given by:
\[ g = \frac{G M}{R_e^2} \]
Therefore, we can express \( G M \) as:
\[ G M = g R_e^2 \]

5. **Substitute \( G M \) into the Potential Energy Formula:**
Now, substituting \( G M \) into the potential energy equation:
\[ U = -\frac{g R_e^2 m}{2R_e} = -\frac{g m R_e}{2} \]

6. **Final Expression for Gravitational Potential Energy:**
Thus, the gravitational potential energy of the body at a height \( R_e \) from the Earth's surface is:
\[ U = -\frac{mg R_e}{2} \]

### Final Answer:
The gravitational potential energy at a height \( R_e \) from the Earth's surface is:
\[ U = -\frac{mg R_e}{2} \]