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 finding the minimum percentage increase in the velocity of a satellite to escape Earth's gravitational pull, we can follow these steps:

### Step 1: Understand the orbital velocity
The orbital velocity \( v_0 \) of a satellite moving around the Earth is given by the formula:
\[ v_0 = \sqrt{gR} \]
where \( g \) is the acceleration due to gravity and \( R \) is the radius of the Earth.

### Step 2: Understand the escape velocity
The escape velocity \( v_e \) is given by the formula:
\[ v_e = \sqrt{2gR} \]

### Step 3: Calculate the increase in velocity
To find the increase in velocity required for the satellite to escape, we subtract the orbital velocity from the escape velocity:
\[ \Delta v = v_e - v_0 = \sqrt{2gR} - \sqrt{gR} \]

### Step 4: Factor out the common term
We can factor out \( \sqrt{gR} \) from the expression:
\[ \Delta v = \sqrt{gR} \left( \sqrt{2} - 1 \right) \]

### Step 5: Calculate the percentage increase
The percentage increase in velocity is given by:
\[ \text{Percentage Increase} = \frac{\Delta v}{v_0} \times 100 \]
Substituting the values we have:
\[ \text{Percentage Increase} = \frac{\sqrt{gR} \left( \sqrt{2} - 1 \right)}{\sqrt{gR}} \times 100 \]
This simplifies to:
\[ \text{Percentage Increase} = \left( \sqrt{2} - 1 \right) \times 100 \]

### Step 6: Calculate the numerical value
We know that \( \sqrt{2} \approx 1.414 \), so:
\[ \sqrt{2} - 1 \approx 0.414 \]
Thus,
\[ \text{Percentage Increase} \approx 0.414 \times 100 \approx 41.4\% \]

### Conclusion
The minimum percentage increase in the satellite's velocity required for it to escape is approximately **41.4%**.

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