How many times is escape velocity `(V_(e))` , of orbital velocity `(V_(0))` for a satellite revolving near earth

Options

A. `sqrt(2)`
B. 2
C. 4
D. `2sqrt(2)`

Answer

To find how many times the escape velocity \( V_e \) is compared to the orbital velocity \( V_o \) for a satellite revolving near Earth, we can follow these steps:

### Step-by-Step Solution:

1. **Write the formula for escape velocity**:
The escape velocity \( V_e \) is given by the formula:
\[ V_e = \sqrt{\frac{2GM}{R}} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the Earth, and \( R \) is the radius of the Earth.

2. **Write the formula for orbital velocity**:
The orbital velocity \( V_o \) is given by the formula:
\[ V_o = \sqrt{\frac{GM}{R}} \]

3. **Set up the ratio of escape velocity to orbital velocity**:
To find how many times the escape velocity is compared to the orbital velocity, we take the ratio:
\[ \frac{V_e}{V_o} = \frac{\sqrt{\frac{2GM}{R}}}{\sqrt{\frac{GM}{R}}} \]

4. **Simplify the ratio**:
We can simplify the expression:
\[ \frac{V_e}{V_o} = \frac{\sqrt{2GM/R}}{\sqrt{GM/R}} = \sqrt{2} \]

5. **Conclusion**:
Therefore, the escape velocity \( V_e \) is \( \sqrt{2} \) times the orbital velocity \( V_o \):
\[ V_e = \sqrt{2} V_o \]

### Final Answer:
The escape velocity \( V_e \) is \( \sqrt{2} \) times the orbital velocity \( V_o \).