What should be the percentage increase in the orbital velcoity to escape velocity ?

Explanation

To determine the percentage increase in orbital velocity to escape velocity, we can follow these steps:

### Step 1: Understand the formulas
The formulas for escape velocity (\(v_e\)) and orbital velocity (\(v_o\)) are:
- Escape velocity: \(v_e = \sqrt{2gR}\)
- Orbital velocity: \(v_o = \sqrt{gR}\)

Where:
- \(g\) is the acceleration due to gravity,
- \(R\) is the radius of the Earth.

### Step 2: Calculate the difference between escape velocity and orbital velocity
To find the increase in velocity, we need to subtract the orbital velocity from the escape velocity:
\[ \Delta v = v_e - v_o = \sqrt{2gR} - \sqrt{gR} \]

### Step 3: Factor out the common term
We can factor out \(\sqrt{gR}\) from the expression:
\[ \Delta v = \sqrt{gR}(\sqrt{2} - 1) \]

### Step 4: Calculate the percentage increase
The percentage increase in velocity is given by the formula:
\[ \text{Percentage Increase} = \left(\frac{\Delta v}{v_o}\right) \times 100 \]
Substituting the values we have:
\[ \text{Percentage Increase} = \left(\frac{\sqrt{gR}(\sqrt{2} - 1)}{\sqrt{gR}}\right) \times 100 \]

### Step 5: Simplify the expression
The \(\sqrt{gR}\) terms cancel out:
\[ \text{Percentage Increase} = (\sqrt{2} - 1) \times 100 \]

### Step 6: Calculate the numerical value
Now we calculate \(\sqrt{2} - 1\):
\[ \sqrt{2} \approx 1.414 \implies \sqrt{2} - 1 \approx 0.414 \]
Thus,
\[ \text{Percentage Increase} \approx 0.414 \times 100 \approx 41.4\% \]

### Final Answer
The percentage increase in orbital velocity to escape velocity is approximately **41.4%**.

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