At what height h above the earth's surface, the value of g becomes g/2 (where R is the radius of the earth)

Options

A. `(sqrt2 - 1) R`
B. `sqrt(2R)`
C. `(sqrt2 + 1) R`
D. `R sqrt2`

Explanation

To find the height \( h \) above the Earth's surface where the acceleration due to gravity \( g \) becomes \( \frac{g}{2} \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
We need to determine the height \( h \) above the Earth's surface where the gravitational acceleration \( g_h \) is equal to \( \frac{g}{2} \). Here, \( R \) is the radius of the Earth.

2. **Using the Formula for Gravitational Acceleration**:
The formula for gravitational acceleration at a height \( h \) above the Earth's surface is given by:
\[ g_h = \frac{g R^2}{(R + h)^2} \]
where \( g \) is the acceleration due to gravity at the Earth's surface.

3. **Setting Up the Equation**:
We want to find \( h \) such that:
\[ g_h = \frac{g}{2} \]
Substituting this into the formula gives:
\[ \frac{g R^2}{(R + h)^2} = \frac{g}{2} \]

4. **Cancelling \( g \)**:
Since \( g \) is present on both sides of the equation, we can cancel it out (assuming \( g \neq 0 \)):
\[ \frac{R^2}{(R + h)^2} = \frac{1}{2} \]

5. **Cross-Multiplying**:
Cross-multiplying gives:
\[ 2R^2 = (R + h)^2 \]

6. **Expanding the Right Side**:
Expanding the right side:
\[ 2R^2 = R^2 + 2Rh + h^2 \]

7. **Rearranging the Equation**:
Rearranging the equation gives:
\[ 2R^2 - R^2 = 2Rh + h^2 \]
\[ R^2 = 2Rh + h^2 \]

8. **Rearranging Further**:
Rearranging further, we have:
\[ h^2 + 2Rh - R^2 = 0 \]

9. **Using the Quadratic Formula**:
This is a quadratic equation in \( h \). We can use the quadratic formula:
\[ h = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 1, b = 2R, c = -R^2 \):
\[ h = \frac{-2R \pm \sqrt{(2R)^2 - 4 \cdot 1 \cdot (-R^2)}}{2 \cdot 1} \]
\[ h = \frac{-2R \pm \sqrt{4R^2 + 4R^2}}{2} \]
\[ h = \frac{-2R \pm \sqrt{8R^2}}{2} \]
\[ h = \frac{-2R \pm 2R\sqrt{2}}{2} \]
\[ h = -R + R\sqrt{2} \]
\[ h = R(\sqrt{2} - 1) \]

10. **Final Result**:
The height \( h \) above the Earth's surface where \( g \) becomes \( \frac{g}{2} \) is:
\[ h = R(\sqrt{2} - 1) \]