The radius in kilometers to which the present radius of the earth (R=6400 km) is to be compressed so that the escape velocity is increases to ten times is

Options

A. 6.4
B. 64
C. 640
D. 4800

Answer

To solve the problem of finding the new radius of the Earth (R') that would increase the escape velocity to ten times the current escape velocity, we can follow these steps:

### Step 1: Write down the formula for escape velocity
The escape velocity (v) from the surface of a planet is given by the formula:
\[ v = \sqrt{\frac{2GM}{R}} \]
where:
- \( G \) is the gravitational constant,
- \( M \) is the mass of the Earth,
- \( R \) is the radius of the Earth.

### Step 2: Set up the equation for the new escape velocity
We want the new escape velocity (v') to be ten times the current escape velocity:
\[ v' = 10v \]
Substituting the formula for escape velocity, we get:
\[ v' = 10 \sqrt{\frac{2GM}{R}} \]

### Step 3: Write the escape velocity for the new radius
The escape velocity for the new radius (R') will be:
\[ v' = \sqrt{\frac{2GM}{R'}} \]

### Step 4: Set the two expressions for escape velocity equal
Since both expressions represent the new escape velocity, we can set them equal to each other:
\[ \sqrt{\frac{2GM}{R'}} = 10 \sqrt{\frac{2GM}{R}} \]

### Step 5: Square both sides to eliminate the square root
Squaring both sides gives:
\[ \frac{2GM}{R'} = 100 \cdot \frac{2GM}{R} \]

### Step 6: Simplify the equation
We can cancel \( 2GM \) from both sides (assuming it is non-zero):
\[ \frac{1}{R'} = \frac{100}{R} \]

### Step 7: Solve for the new radius (R')
Rearranging gives:
\[ R' = \frac{R}{100} \]

### Step 8: Substitute the current radius of the Earth
Given that the current radius \( R = 6400 \) km, we substitute this value:
\[ R' = \frac{6400 \text{ km}}{100} = 64 \text{ km} \]

### Final Answer
The radius to which the Earth must be compressed so that the escape velocity increases to ten times is:
\[ R' = 64 \text{ km} \]

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