Consider two satellites `A` and `B` of equal mass, moving in the same circular orbit of radius `r` around the earth but in the opposite sense and therefore a collision occurs.
(a) Find the total mechanical energy `E_(A) + E_(B)` of the two satellite-plus-earth system before collision.
(b) If the collision is completely inelastic, find the total mechanical energy immediately after collision. Describe the subsequent motion of the combined satellite.

Explanation

To solve the problem step by step, we will address both parts of the question regarding the two satellites A and B.

### (a) Total Mechanical Energy Before Collision

1. **Identify the Orbital Velocity**:
The orbital velocity \( v \) of a satellite in a circular orbit of radius \( r \) around the Earth (mass \( M \)) is given by the formula:
\[ v = \sqrt{\frac{GM}{r}} \]
where \( G \) is the gravitational constant.

2. **Calculate Kinetic Energy of Each Satellite**:
The kinetic energy \( K \) of each satellite (mass \( m \)) is:
\[ K_A = \frac{1}{2} m v^2 = \frac{1}{2} m \left(\frac{GM}{r}\right) = \frac{GMm}{2r} \]
Since both satellites have the same mass and are moving with the same speed, the kinetic energy of satellite B will be the same:
\[ K_B = \frac{1}{2} m v^2 = \frac{GMm}{2r} \]

3. **Calculate Potential Energy of Each Satellite**:
The gravitational potential energy \( U \) of each satellite is given by:
\[ U_A = U_B = -\frac{GMm}{r} \]

4. **Total Mechanical Energy Before Collision**:
The total mechanical energy \( E \) of the two satellites plus Earth before the collision is:
\[ E_{A} + E_{B} = (K_A + U_A) + (K_B + U_B) \]
Substituting the values:
\[ E_{A} + E_{B} = \left(\frac{GMm}{2r} - \frac{GMm}{r}\right) + \left(\frac{GMm}{2r} - \frac{GMm}{r}\right) \]
Simplifying:
\[ E_{A} + E_{B} = \left(\frac{GMm}{2r} + \frac{GMm}{2r}\right) - \left(\frac{GMm}{r} + \frac{GMm}{r}\right) = -\frac{GMm}{r} \]

### (b) Total Mechanical Energy Immediately After Collision

1. **Conservation of Linear Momentum**:
Since the collision is completely inelastic, both satellites stick together after the collision. The initial momentum before collision is:
\[ p_{\text{initial}} = m \cdot v + m \cdot (-v) = 0 \]
Therefore, the momentum after collision is also zero, meaning the combined mass \( 2m \) will have a velocity \( V' = 0 \).

2. **Total Mechanical Energy After Collision**:
After the collision, the kinetic energy of the combined mass is:
\[ K' = \frac{1}{2} (2m) (0)^2 = 0 \]
The potential energy of the combined mass in the same orbit is:
\[ U' = -\frac{GM(2m)}{r} = -\frac{2GMm}{r} \]

3. **Total Mechanical Energy After Collision**:
The total mechanical energy after the collision is:
\[ E' = K' + U' = 0 - \frac{2GMm}{r} = -\frac{2GMm}{r} \]

4. **Subsequent Motion of the Combined Satellite**:
After the collision, since the combined satellite has zero velocity, it will begin to fall towards the Earth due to gravity. As it falls, it will lose potential energy and gain kinetic energy until it impacts the Earth.

### Summary of Solutions
- **Total Mechanical Energy Before Collision**:
\[ E_{A} + E_{B} = -\frac{GMm}{r} \]
- **Total Mechanical Energy After Collision**:
\[ E' = -\frac{2GMm}{r} \]
- **Subsequent Motion**: The combined satellite will fall towards the Earth.