A man weights 60 kg at earth's surface. At what height above the earth's surface weight becomes 30 kg. Given radius of earth is 6400 km.

Options

A. 2624 km
B. 3000 km
C. 2020 km
D. None of these

Answer

To solve the problem of finding the height above the Earth's surface where a man's weight becomes 30 kg, we can follow these steps:

### Step 1: Understand the relationship between weight and gravity
Weight (W) is given by the formula:
\[ W = m \cdot g \]
where:
- \( W \) is the weight,
- \( m \) is the mass (which remains constant),
- \( g \) is the acceleration due to gravity.

At the Earth's surface, the man weighs 60 kg, which corresponds to a weight of:
\[ W = 60 \, \text{kg} \cdot g \]

### Step 2: Determine the weight at the new height
We want to find the height where the weight becomes 30 kg. Using the same formula:
\[ W' = m \cdot g' \]
where \( W' = 30 \, \text{kg} \) and \( g' \) is the new acceleration due to gravity at height \( h \).

### Step 3: Set up the equation
Since the mass \( m \) is constant, we can express the relationship between the weights:
\[ \frac{W'}{W} = \frac{g'}{g} \]
Substituting the known values:
\[ \frac{30}{60} = \frac{g'}{g} \]
This simplifies to:
\[ g' = \frac{g}{2} \]

### Step 4: Use the formula for gravity at height
The formula for gravity at a distance \( r + h \) from the center of the Earth is:
\[ g' = \frac{G \cdot M}{(R + h)^2} \]
where:
- \( G \) is the gravitational constant,
- \( M \) is the mass of the Earth,
- \( R \) is the radius of the Earth (6400 km).

At the surface of the Earth, the gravity is:
\[ g = \frac{G \cdot M}{R^2} \]

### Step 5: Set up the equation for \( g' \)
Since we know \( g' = \frac{g}{2} \), we can write:
\[ \frac{g}{2} = \frac{G \cdot M}{(R + h)^2} \]

### Step 6: Substitute \( g \)
Substituting \( g \) into the equation:
\[ \frac{G \cdot M}{2} = \frac{G \cdot M}{(R + h)^2} \]
Cancelling \( G \cdot M \) from both sides gives:
\[ \frac{1}{2} = \frac{1}{(R + h)^2} \]

### Step 7: Rearrange the equation
Rearranging the equation gives:
\[ (R + h)^2 = 2R^2 \]

### Step 8: Solve for \( h \)
Taking the square root of both sides:
\[ R + h = R\sqrt{2} \]
Thus:
\[ h = R\sqrt{2} - R \]
\[ h = R(\sqrt{2} - 1) \]

### Step 9: Substitute the radius of the Earth
Substituting \( R = 6400 \, \text{km} \):
\[ h = 6400 \cdot (\sqrt{2} - 1) \]
Calculating \( \sqrt{2} \approx 1.414 \):
\[ h = 6400 \cdot (1.414 - 1) \]
\[ h = 6400 \cdot 0.414 \]
\[ h \approx 2649.6 \, \text{km} \]

### Final Answer
The height above the Earth's surface where the man's weight becomes 30 kg is approximately **2649.6 km**.

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