How far away from the surface of earth does the acceleration due to gravity become 4% of its value on the surface of earth ? `[R_(e )=6400 km]`

Explanation

To find out how far away from the surface of the Earth the acceleration due to gravity becomes 4% of its value on the surface, we can follow these steps:

### Step 1: Understand the formula for acceleration due to gravity at height \( h \)
The acceleration due to gravity at a height \( h \) above the Earth's surface is given by the formula:
\[ g_h = \frac{g}{(1 + \frac{h}{R})^2} \]
where:
- \( g_h \) is the acceleration due to gravity at height \( h \),
- \( g \) is the acceleration due to gravity on the surface of the Earth,
- \( R \) is the radius of the Earth.

### Step 2: Set up the equation for 4% of surface gravity
We need to find \( h \) such that \( g_h = 0.04g \) (which is 4% of \( g \)). Therefore, we can write:
\[ 0.04g = \frac{g}{(1 + \frac{h}{R})^2} \]

### Step 3: Simplify the equation
We can cancel \( g \) from both sides of the equation (assuming \( g \neq 0 \)):
\[ 0.04 = \frac{1}{(1 + \frac{h}{R})^2} \]

### Step 4: Rearranging the equation
Taking the reciprocal of both sides gives:
\[ (1 + \frac{h}{R})^2 = \frac{1}{0.04} \]
Calculating \( \frac{1}{0.04} \):
\[ \frac{1}{0.04} = 25 \]
Thus, we have:
\[ (1 + \frac{h}{R})^2 = 25 \]

### Step 5: Taking the square root
Taking the square root of both sides results in:
\[ 1 + \frac{h}{R} = 5 \]

### Step 6: Solve for \( h \)
Now, we can isolate \( h \):
\[ \frac{h}{R} = 5 - 1 = 4 \]
Multiplying both sides by \( R \):
\[ h = 4R \]

### Step 7: Substitute the value of \( R \)
Given that \( R = 6400 \, \text{km} \):
\[ h = 4 \times 6400 \, \text{km} = 25600 \, \text{km} \]

### Conclusion
The height \( h \) at which the acceleration due to gravity becomes 4% of its value on the surface of the Earth is:
\[ \boxed{25600 \, \text{km}} \]