The height from the surface of earth at which the gravitational potential energy of a ball of mass `m` is half of that at the centre of earth is (where `R` is the radius of earth)

Options

A. (a)`(R)/(4)`
B. (b)`(R)/(3)`
C. (c)`(3R)/(4)`
D. (d)`(4R)/(3)`

Explanation

To solve the problem, we need to find the height \( h \) from the surface of the Earth at which the gravitational potential energy (GPE) of a ball of mass \( m \) is half of that at the center of the Earth.

### Step-by-Step Solution:

1. **Understanding Gravitational Potential Energy (GPE)**:
- The gravitational potential energy \( U \) 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 \( m \) is the mass of the ball.

2. **GPE at the Center of the Earth**:
- At the center of the Earth (where \( r = 0 \)), the potential energy \( U_c \) is given by:
\[ U_c = -\frac{3}{2} \frac{G M m}{R} \]
- Here, \( R \) is the radius of the Earth.

3. **GPE at Height \( h \)**:
- At a height \( h \) above the surface of the Earth, the distance from the center of the Earth becomes \( R + h \). Therefore, the potential energy \( U_a \) at this height is:
\[ U_a = -\frac{G M m}{R + h} \]

4. **Setting Up the Equation**:
- According to the problem, we need to find \( h \) such that:
\[ U_a = \frac{1}{2} U_c \]
- Substituting the expressions for \( U_a \) and \( U_c \):
\[ -\frac{G M m}{R + h} = \frac{1}{2} \left(-\frac{3}{2} \frac{G M m}{R}\right) \]

5. **Simplifying the Equation**:
- Canceling the common terms \( -G M m \) from both sides gives:
\[ \frac{1}{R + h} = \frac{3}{4R} \]

6. **Cross-Multiplying**:
- Cross-multiplying to solve for \( h \):
\[ 4R = 3(R + h) \]
- Expanding the right side:
\[ 4R = 3R + 3h \]

7. **Isolating \( h \)**:
- Rearranging the equation to isolate \( h \):
\[ 4R - 3R = 3h \]
\[ R = 3h \]
\[ h = \frac{R}{3} \]

8. **Conclusion**:
- The height \( h \) from the surface of the Earth at which the gravitational potential energy of the ball is half of that at the center of the Earth is:
\[ h = \frac{R}{3} \]

### Final Answer:
The height from the surface of the Earth is \( \frac{R}{3} \).