Find the height over the eart's surface at which the weight of a body becomes half of its alue t the surface.

Explanation

To find the height above the Earth's surface at which the weight of a body becomes half of its value at the surface, we can follow these steps:

### Step 1: Understand the Weight at the Surface and at Height
The weight of a body at the surface of the Earth is given by the formula:
\[ W = \frac{G \cdot m \cdot M}{R^2} \]
where:
- \( G \) is the gravitational constant,
- \( m \) is the mass of the body,
- \( M \) is the mass of the Earth,
- \( R \) is the radius of the Earth.

At a height \( h \) above the Earth's surface, the weight \( W_h \) is given by:
\[ W_h = \frac{G \cdot m \cdot M}{(R + h)^2} \]

### Step 2: Set Up the Equation
According to the problem, we want the weight at height \( h \) to be half of the weight at the surface:
\[ W_h = \frac{1}{2} W \]
Substituting the expressions for weight, we get:
\[ \frac{G \cdot m \cdot M}{(R + h)^2} = \frac{1}{2} \cdot \frac{G \cdot m \cdot M}{R^2} \]

### Step 3: Simplify the Equation
We can cancel \( G \), \( m \), and \( M \) from both sides:
\[ \frac{1}{(R + h)^2} = \frac{1}{2R^2} \]

### Step 4: Cross-Multiply
Cross-multiplying gives:
\[ 2R^2 = (R + h)^2 \]

### Step 5: Expand the Right Side
Expanding the right side:
\[ 2R^2 = R^2 + 2Rh + h^2 \]

### Step 6: Rearrange the Equation
Rearranging the equation gives:
\[ 2R^2 - R^2 = 2Rh + h^2 \]
\[ R^2 = 2Rh + h^2 \]

### Step 7: Rearranging to Form a Quadratic Equation
Rearranging the equation leads to:
\[ h^2 + 2Rh - R^2 = 0 \]

### Step 8: Solve the Quadratic Equation
Using the quadratic formula \( h = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1, b = 2R, c = -R^2 \):
\[ h = \frac{-2R \pm \sqrt{(2R)^2 - 4 \cdot 1 \cdot (-R^2)}}{2 \cdot 1} \]
\[ h = \frac{-2R \pm \sqrt{4R^2 + 4R^2}}{2} \]
\[ h = \frac{-2R \pm \sqrt{8R^2}}{2} \]
\[ h = \frac{-2R \pm 2R\sqrt{2}}{2} \]
\[ h = -R + R\sqrt{2} \]
\[ h = R(\sqrt{2} - 1) \]

### Final Result
Thus, the height \( h \) at which the weight of the body becomes half of its value at the surface is:
\[ h = R(\sqrt{2} - 1) \]