The relation between the escape velocity `v_e` and orbital velocity `v_0` is given by

Options

A. `V_e=sqrt2V_0`
B. `V_0=sqrt2V_e`
C. `sqrtV_e= 2V_0`
D. `V_e=V_0`

Answer

To derive the relation between escape velocity \( v_e \) and orbital velocity \( v_0 \), we can follow these steps:

### Step 1: Understand the formulas for orbital and escape velocity
The orbital velocity \( v_0 \) for a satellite in a circular orbit around a planet is given by the formula:
\[ v_0 = \sqrt{gR} \]
where \( g \) is the acceleration due to gravity at the surface of the planet and \( R \) is the radius of the planet.

### Step 2: Write the formula for escape velocity
The escape velocity \( v_e \) from the surface of a planet is given by:
\[ v_e = \sqrt{2gR} \]

### Step 3: Set up the ratio of escape velocity to orbital velocity
To find the relationship between \( v_e \) and \( v_0 \), we can divide the escape velocity by the orbital velocity:
\[ \frac{v_e}{v_0} = \frac{\sqrt{2gR}}{\sqrt{gR}} \]

### Step 4: Simplify the ratio
This simplifies as follows:
\[ \frac{v_e}{v_0} = \frac{\sqrt{2}}{\sqrt{1}} = \sqrt{2} \]

### Step 5: Conclude the relationship
From the above, we can express the escape velocity in terms of the orbital velocity:
\[ v_e = \sqrt{2} \cdot v_0 \]

### Final Answer
The relation between escape velocity \( v_e \) and orbital velocity \( v_0 \) is:
\[ v_e = \sqrt{2} \cdot v_0 \]

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