At a certain height above the earth's surface, the acceleration due to gravity is 4% of its value at the surface of the earth. Determine the height.

Options

A. 2R
B. 4R
C. 3R
D. `R//2`

Answer

To solve the problem, we need to determine the height above the Earth's surface where the acceleration due to gravity is 4% of its value at the surface. Let's follow the steps to find the solution.

### Step-by-Step Solution:

1. **Understand the relationship between gravity at height and at the surface:**
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 \( g \) is the acceleration due to gravity at the surface, and \( R \) is the radius of the Earth.

2. **Set up the equation based on the problem statement:**
According to the problem, the acceleration due to gravity at height \( h \) is 4% of that at the surface:
\[ g_h = 0.04g \]

3. **Substitute \( g_h \) into the equation:**
We can substitute \( g_h \) from the first equation into the second:
\[ \frac{g}{(1 + \frac{h}{R})^2} = 0.04g \]

4. **Cancel \( g \) from both sides (assuming \( g \neq 0 \)):**
\[ \frac{1}{(1 + \frac{h}{R})^2} = 0.04 \]

5. **Take the reciprocal of both sides:**
\[ (1 + \frac{h}{R})^2 = \frac{1}{0.04} \]
Simplifying \( \frac{1}{0.04} \):
\[ (1 + \frac{h}{R})^2 = 25 \]

6. **Take the square root of both sides:**
\[ 1 + \frac{h}{R} = 5 \]

7. **Solve for \( \frac{h}{R} \):**
\[ \frac{h}{R} = 5 - 1 \]
\[ \frac{h}{R} = 4 \]

8. **Finally, solve for \( h \):**
\[ h = 4R \]

### Conclusion:
The height \( h \) above the Earth's surface where the acceleration due to gravity is 4% of its value at the surface is:
\[ h = 4R \]
where \( R \) is the radius of the Earth.