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)` from the earth's surface will be (here `R_(e)` is the radius of the earth)

Options

A. `mgR`
B. `0.67mgR`
C. `-(mgR)/2`
D. `0.33mgR`

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

### Step 2: Determine the distance from the center of the Earth at height Re
At the Earth's surface, the distance from the center of the Earth is equal to the radius of the Earth (Re). When the body is raised to a height equal to the radius of the Earth (Re), the total distance from the center of the Earth becomes:
\[ r = R_e + R_e = 2R_e \]

### Step 3: Substitute the values into the potential energy formula
Now, substituting \( r = 2R_e \) into the gravitational potential energy formula:
\[ U = -\frac{GMm}{2R_e} \]

### Step 4: Relate GM to mg
We know that the gravitational force at the surface of the Earth is given by:
\[ mg = \frac{GMm}{R_e^2} \]
From this, we can express \( GM \) as:
\[ GM = gR_e^2 \]
Now substituting this into the potential energy formula:
\[ U = -\frac{gR_e^2 m}{2R_e} \]

### Step 5: Simplify the expression
This simplifies to:
\[ U = -\frac{gRm}{2} \]

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

### Final Answer
The gravitational potential energy at a height \( R_e \) from the Earth's surface is:
\[ U = -\frac{1}{2} mgR_e \]

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