A satellite is orbiting the earth close to its surface. A particle is to be projected from the satellite to just escape from the earth. The escape speed from the earth is `v_e`. Its speed with respect to the satellite

Options

A. wil be less than `v_e`
B. will be more thant `v_e`
C. will be equal to `v_e`
D. will depend on direction of projection

Answer

To solve the problem, we need to analyze the situation where a satellite is orbiting the Earth close to its surface and a particle is projected from the satellite to escape the Earth's gravitational pull.

### Step-by-Step Solution:

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

2. **Velocity of the Satellite**:
The satellite, being in orbit close to the Earth's surface, has a certain orbital velocity \( v_s \). For a satellite in a low Earth orbit, this velocity can be approximated as:
\[ v_s \approx \sqrt{\frac{GM}{R}} = v_e \]
Thus, the speed of the satellite \( v_s \) is approximately equal to the escape velocity \( v_e \).

3. **Relative Velocity**:
When the particle is projected from the satellite, we denote its velocity as \( v_p \). The velocity of the particle with respect to the satellite is given by:
\[ v_{p,s} = v_p - v_s \]
where \( v_{p,s} \) is the speed of the particle with respect to the satellite.

4. **Direction of Projection**:
The direction in which the particle is projected affects its speed relative to the satellite. If the particle is projected in the same direction as the satellite's motion, it will need less speed to reach escape velocity. Conversely, if projected in the opposite direction, it will need more speed.

5. **Conclusion**:
Since the escape velocity from the Earth is \( v_e \) and the satellite's speed is also approximately \( v_e \), the speed of the particle with respect to the satellite will depend on the direction of projection. Therefore, the answer is that the speed of the particle with respect to the satellite will depend on the direction of projection.

### Final Answer:
The speed of the particle with respect to the satellite will depend on the direction of projection. Thus, the correct option is **D**.