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. R
D. R/2

Answer

To solve the problem of determining the height above the Earth's surface where the acceleration due to gravity is 4% of its value at the surface, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the formula for acceleration due to gravity at a height**:
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 at the surface of the Earth,
- \( R \) is the radius of the Earth.

2. **Set up the equation based on the given information**:
According to the problem, the acceleration due to gravity at height \( h \) is 4% of its value at the surface. Therefore, we can express this as:
\[ g_h = 0.04g \]

3. **Substitute \( g_h \) into the formula**:
Now we substitute \( g_h \) into the equation:
\[ 0.04g = \frac{g}{(1 + \frac{h}{R})^2} \]

4. **Cancel \( g \) from both sides**:
Since \( g \) is common on both sides (and not equal to zero), we can cancel it out:
\[ 0.04 = \frac{1}{(1 + \frac{h}{R})^2} \]

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

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

7. **Solve for \( \frac{h}{R} \)**:
Subtracting 1 from both sides gives:
\[ \frac{h}{R} = 4 \]

8. **Calculate the height \( h \)**:
Multiplying both sides by \( R \) results in:
\[ h = 4R \]

### Final Answer:
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.