A rocket is fired from inside a deep mine, so as to escape the earth’s gravitational field. The minimum velocity to be imparted to the rocket is

Options

A. Exactly the same as the escape velocity of fire from the earth’s surface
B. A little more than the escape velocity of fire from the earth’s surface
C. A little less than the escape velocity of fire from the earth’s surface
D. Infinity

Explanation

To find the minimum velocity required for a rocket to escape the Earth's gravitational field when fired from inside a deep mine, we need to understand the concept of escape velocity.

### Step-by-Step Solution:

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

2. **Considering the Depth of the Mine**:
When a rocket is fired from a depth \( h \) below the Earth's surface, the effective gravitational force it experiences changes. The distance from the center of the Earth to the rocket is now \( R - h \).

3. **Escape Velocity from Depth**:
The escape velocity from a depth \( h \) can be derived from the formula for escape velocity, but we need to consider the gravitational force acting at that depth. The formula for escape velocity from a depth \( h \) becomes:
\[ v' = \sqrt{\frac{2GM}{R - h}} \]
This indicates that the escape velocity from a depth is dependent on the distance from the center of the Earth to the rocket.

4. **Comparing Escape Velocities**:
Since \( R - h < R \), it follows that:
\[ \frac{2GM}{R - h} > \frac{2GM}{R} \]
Therefore, the escape velocity from a depth \( h \) is greater than the escape velocity from the surface of the Earth.

5. **Conclusion**:
Thus, the minimum velocity required for the rocket to escape the Earth's gravitational field when fired from inside a deep mine is **a little more than the escape velocity from the surface of the Earth**.

### Final Answer:
The minimum velocity to be imparted to the rocket is **a little more than the escape velocity of the rocket fired from the Earth's surface**.