When a satellite is moving around the earth with velocity `v`, then to make the satellite escape, the minimum percentage increase in its velocity should be

Options

A. `100 %`
B. `82.4 %`
C. `41.4 %`
D. None of these

Explanation

To solve the problem of determining the minimum percentage increase in the velocity of a satellite moving around the Earth in order to escape, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Current Velocity of the Satellite**:
The velocity \( v \) of a satellite in a circular orbit around the Earth is given by:
\[ v = \sqrt{g \cdot r} \]
where \( g \) is the acceleration due to gravity and \( r \) is the distance from the center of the Earth to the satellite.

2. **Determine the Escape Velocity**:
The escape velocity \( v_e \) required for the satellite to escape the gravitational pull of the Earth is given by:
\[ v_e = \sqrt{2g \cdot r} \]

3. **Calculate the Change in Velocity**:
The change in velocity \( \Delta v \) needed for the satellite to escape can be calculated as:
\[ \Delta v = v_e - v \]
Substituting the expressions for \( v_e \) and \( v \):
\[ \Delta v = \sqrt{2g \cdot r} - \sqrt{g \cdot r} \]

4. **Factor Out Common Terms**:
We can factor out \( \sqrt{g \cdot r} \) from the equation:
\[ \Delta v = \sqrt{g \cdot r} \left( \sqrt{2} - 1 \right) \]

5. **Calculate the Value of \( \sqrt{2} - 1 \)**:
We know that \( \sqrt{2} \approx 1.414 \), thus:
\[ \sqrt{2} - 1 \approx 0.414 \]
Therefore, we can express \( \Delta v \) as:
\[ \Delta v \approx 0.414 \sqrt{g \cdot r} \]

6. **Calculate the Percentage Increase in Velocity**:
The percentage increase in velocity can be calculated using the formula:
\[ \text{Percentage Increase} = \left( \frac{\Delta v}{v} \right) \times 100 \]
Substituting the values of \( \Delta v \) and \( v \):
\[ \text{Percentage Increase} = \left( \frac{0.414 \sqrt{g \cdot r}}{\sqrt{g \cdot r}} \right) \times 100 \]
This simplifies to:
\[ \text{Percentage Increase} = 0.414 \times 100 = 41.4\% \]

### Final Answer:
The minimum percentage increase in the velocity of the satellite to make it escape is **41.4%**.