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 need to analyze the situation before and after the collision between the satellite and the meteorite.

### Step-by-Step Solution:

1. **Identify the Initial Conditions**:
- The satellite has a mass \( M \) and is in a circular orbit of radius \( R \) around the Earth.
- The speed of the satellite in its circular orbit can be calculated using the formula for orbital velocity:
\[ v_0 = \sqrt{\frac{GM}{R}} \]
- The meteorite, also of mass \( M \), is falling towards the Earth and has the same speed \( v_0 \) just before the collision.

2. **Analyze the Collision**:
- The collision between the satellite and the meteorite is completely inelastic, meaning they stick together after the collision.
- Before the collision, the velocities of the satellite and meteorite are:
- Satellite: \( \vec{v}_{satellite} = v_0 \hat{i} \) (assuming it moves horizontally)
- Meteorite: \( \vec{v}_{meteorite} = -v_0 \hat{j} \) (falling vertically)

3. **Apply Conservation of Momentum**:
- The total momentum before the collision can be expressed as:
\[ \vec{p}_{initial} = M \vec{v}_{satellite} + M \vec{v}_{meteorite} = M v_0 \hat{i} + M (-v_0 \hat{j}) = M v_0 \hat{i} - M v_0 \hat{j} \]
- After the collision, the combined mass is \( 2M \) and let the velocity of the combined body be \( \vec{v} \).
- By conservation of momentum:
\[ \vec{p}_{initial} = \vec{p}_{final} \]
\[ M v_0 \hat{i} - M v_0 \hat{j} = 2M \vec{v} \]
- Dividing through by \( M \):
\[ v_0 \hat{i} - v_0 \hat{j} = 2 \vec{v} \]
- Therefore:
\[ \vec{v} = \frac{1}{2}(v_0 \hat{i} - v_0 \hat{j}) = \frac{v_0}{2} \hat{i} - \frac{v_0}{2} \hat{j} \]

4. **Calculate the Magnitude of the Velocity**:
- The magnitude of the velocity \( \vec{v} \) is:
\[ |\vec{v}| = \sqrt{\left(\frac{v_0}{2}\right)^2 + \left(-\frac{v_0}{2}\right)^2} = \sqrt{\frac{v_0^2}{4} + \frac{v_0^2}{4}} = \sqrt{\frac{v_0^2}{2}} = \frac{v_0}{\sqrt{2}} \]

5. **Determine the Subsequent Motion**:
- Since the new speed \( |\vec{v}| = \frac{v_0}{\sqrt{2}} \) is less than the orbital speed \( v_0 \), the combined body will not have enough speed to maintain a circular orbit.
- Instead, it will enter an elliptical orbit around the Earth.

### Conclusion:
The subsequent motion of the combined body after the collision will be an elliptical orbit around the Earth.