A body weight `W`,is projected vertically upwards from earth's surface to reach a height above the earth which is equal to nine times the radius of earth.The weight of the body at that height will be:

Options

A. `(W)/(100)`
B. `(W)/(3)`
C. `(W)/(91)`
D. `(W)/(9)`

Answer

To solve the problem, we need to determine the weight of a body at a height above the Earth's surface that is equal to nine times the radius of the Earth. Let's break this down step by step.

### Step 1: Understand the Weight Formula
The weight of a body at a distance \( r \) from the center of the Earth is given by the formula:
\[ W' = \frac{G \cdot M \cdot m}{r^2} \]
where:
- \( W' \) is the weight of the body at distance \( r \),
- \( G \) is the universal gravitational constant,
- \( M \) is the mass of the Earth,
- \( m \) is the mass of the body,
- \( r \) is the distance from the center of the Earth.

### Step 2: Determine the Distance from the Center of the Earth
The radius of the Earth is denoted as \( R_e \). The height above the Earth's surface is given as \( 9R_e \). Therefore, the total distance from the center of the Earth when the body is at this height is:
\[ r = R_e + 9R_e = 10R_e \]

### Step 3: Substitute into the Weight Formula
Now, we can substitute \( r = 10R_e \) into the weight formula:
\[ W' = \frac{G \cdot M \cdot m}{(10R_e)^2} \]
This simplifies to:
\[ W' = \frac{G \cdot M \cdot m}{100R_e^2} \]

### Step 4: Relate to Weight at Earth's Surface
The weight of the body at the Earth's surface is given by:
\[ W = \frac{G \cdot M \cdot m}{R_e^2} \]
Now, we can relate \( W' \) to \( W \):
\[ W' = \frac{1}{100} \cdot \frac{G \cdot M \cdot m}{R_e^2} = \frac{1}{100} W \]

### Step 5: Conclusion
Thus, the weight of the body at the height of \( 9R_e \) above the Earth's surface is:
\[ W' = \frac{W}{100} \]

### Final Answer
The weight of the body at that height will be \( \frac{W}{100} \).

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