At what height above the surface of the earth will the acceleration due to gravity be `25%` of its value on the surface of the earth ? Assume that the radius of the earth is 6400 km .

Answer

To find the height above the surface of the Earth where the acceleration due to gravity is 25% of its value on the surface, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the formula for acceleration due to gravity (g)**:
The acceleration due to gravity at a distance \( r \) from the center of the Earth is given by:
\[ g' = \frac{GM}{r^2} \]
where \( G \) is the gravitational constant and \( M \) is the mass of the Earth.

2. **Establish the relationship for 25% of surface gravity**:
The acceleration due to gravity at the surface of the Earth is:
\[ g = \frac{GM}{R^2} \]
where \( R \) is the radius of the Earth (6400 km). We want to find the height \( h \) such that:
\[ g' = \frac{g}{4} \]

3. **Set up the equation**:
At height \( h \) above the surface, the distance from the center of the Earth becomes \( R + h \). Therefore, we can write:
\[ g' = \frac{GM}{(R + h)^2} \]
Setting this equal to \( \frac{g}{4} \):
\[ \frac{GM}{(R + h)^2} = \frac{1}{4} \cdot \frac{GM}{R^2} \]

4. **Cancel \( GM \) from both sides**:
Since \( GM \) is common on both sides, we can simplify:
\[ \frac{1}{(R + h)^2} = \frac{1}{4R^2} \]

5. **Cross-multiply to solve for \( R + h \)**:
\[ 4R^2 = (R + h)^2 \]

6. **Expand the right-hand side**:
\[ 4R^2 = R^2 + 2Rh + h^2 \]

7. **Rearrange the equation**:
\[ 4R^2 - R^2 = 2Rh + h^2 \]
\[ 3R^2 = 2Rh + h^2 \]

8. **Rearranging gives us a quadratic equation**:
\[ h^2 + 2Rh - 3R^2 = 0 \]

9. **Use the quadratic formula to solve for \( h \)**:
The quadratic formula is given by:
\[ h = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1 \), \( b = 2R \), and \( c = -3R^2 \):
\[ h = \frac{-2R \pm \sqrt{(2R)^2 - 4 \cdot 1 \cdot (-3R^2)}}{2 \cdot 1} \]
\[ h = \frac{-2R \pm \sqrt{4R^2 + 12R^2}}{2} \]
\[ h = \frac{-2R \pm \sqrt{16R^2}}{2} \]
\[ h = \frac{-2R \pm 4R}{2} \]

10. **Calculate the possible values for \( h \)**:
- Taking the positive root:
\[ h = \frac{2R}{2} = R \]
- Since \( R = 6400 \text{ km} \):
\[ h = 6400 \text{ km} \]

### Final Answer:
The height above the surface of the Earth where the acceleration due to gravity is 25% of its value on the surface is **6400 km**.