A satellite of mass M is in a circular orbit of radius R about the centre of the earth. A meteorite of the same mass, falling towards the earth, collides with the satellite completely inelastically. The speeds of the satellite and the meteorite are the same, just before the collision. The subsequent motion of the combined body will be:

Options

A. In the same circular orbit of radius R
B. In a elliptical orbit
C. Such that it escapes to infinity
D. In a circular orbit of a different radius

Explanation

To solve the problem, we will analyze the situation step by step, applying the principles of conservation of momentum and understanding the implications of the collision.

### Step-by-Step Solution:

1. **Identify the Given Information:**
- Mass of the satellite, \( M \).
- Mass of the meteorite, \( M \) (same as the satellite).
- Radius of the satellite's orbit, \( R \).
- The speed of the satellite in a circular orbit, \( v_s = \sqrt{\frac{GM}{R}} \).
- The speed of the meteorite just before the collision, \( v_m = v_s \) (since they are the same).

2. **Determine the Initial Velocities:**
- The satellite is moving in a circular orbit with velocity \( v_s \).
- The meteorite is falling towards the Earth, and its velocity just before the collision is also \( v_s \).

3. **Set Up the Conservation of Momentum:**
- Before the collision, the momentum of the satellite is \( M v_s \) (in the tangential direction).
- The momentum of the meteorite is \( M v_s \) (in the radial direction towards the Earth).
- Since the collision is completely inelastic, they stick together after the collision.

4. **Calculate the Total Momentum Before Collision:**
- The total momentum vector before the collision can be expressed as:
\[ \text{Total Momentum} = M v_s \hat{j} + M (-v_s \hat{i}) = M v_s (\hat{j} - \hat{i}) \]

5. **Calculate the Combined Mass After Collision:**
- After the collision, the combined mass is \( 2M \).

6. **Apply Conservation of Momentum:**
- The momentum after the collision must equal the momentum before the collision:
\[ 2M \vec{v} = M v_s (\hat{j} - \hat{i}) \]
- Dividing both sides by \( 2M \):
\[ \vec{v} = \frac{1}{2} v_s (\hat{j} - \hat{i}) \]

7. **Magnitude of the Velocity After Collision:**
- The magnitude of the velocity \( v \) is:
\[ v = \frac{v_s}{2} \sqrt{2} = \frac{v_s}{\sqrt{2}} \]

8. **Determine the Nature of the Orbit:**
- Since the new speed \( v \) is less than the original orbital speed \( v_s \), the combined body will not maintain a circular orbit.
- Instead, it will move in an elliptical orbit, as the speed is now lower than the required speed for a circular orbit.

### Conclusion:
The subsequent motion of the combined body (satellite + meteorite) will be elliptical.