A man weighs 60kg on the surface of the earth. The altitude where his weight will be 30kg is (R is the radius of the earth)

Options

A. `h=2sqrtR`
B. h=2R
C. `h=R/2`
D. `h=(sqrt2-1)R`

Answer

To solve the problem, we need to determine the altitude at which a man weighing 60 kg on the surface of the Earth will weigh only 30 kg. We will use the relationship between weight, mass, and gravitational acceleration.

### Step-by-Step Solution:

1. **Understanding Weight on the Surface of the Earth**:
- The weight of the man on the surface of the Earth is given by the formula:
\[ W = m \cdot g \]
- Here, \( m = 60 \, \text{kg} \) (mass of the man) and \( g \) is the acceleration due to gravity at the surface of the Earth.
- Therefore, the weight on the surface is:
\[ W = 60g \]

2. **Weight at Height \( h \)**:
- At a height \( h \) above the Earth's surface, the weight of the man is given as 30 kg. Thus, we can express this as:
\[ W_h = m \cdot g_h = 30g \]
- Here, \( g_h \) is the acceleration due to gravity at height \( h \).

3. **Using the Formula for Gravitational Acceleration**:
- The gravitational acceleration at a height \( h \) can be expressed as:
\[ g_h = \frac{g \cdot R^2}{(R + h)^2} \]
- Where \( R \) is the radius of the Earth.

4. **Setting Up the Equation**:
- We can set up the equation based on the weights:
\[ 30g = 60 \cdot \frac{g \cdot R^2}{(R + h)^2} \]
- Dividing both sides by \( g \) (assuming \( g \neq 0 \)):
\[ 30 = 60 \cdot \frac{R^2}{(R + h)^2} \]

5. **Simplifying the Equation**:
- Dividing both sides by 30:
\[ 1 = 2 \cdot \frac{R^2}{(R + h)^2} \]
- Rearranging gives:
\[ (R + h)^2 = 2R^2 \]

6. **Taking the Square Root**:
- Taking the square root of both sides:
\[ R + h = R\sqrt{2} \]

7. **Solving for \( h \)**:
- Rearranging to find \( h \):
\[ h = R\sqrt{2} - R = R(\sqrt{2} - 1) \]

### Final Answer:
The altitude \( h \) where the man's weight will be 30 kg is:
\[ h = R(\sqrt{2} - 1) \]