Two satellite of same mass are launched in the same orbit of radius `r` around the earth so as to rotate opposite to each other. If they collide inelastically and stick together as wreckage, the total energy of the system just after collision is

Options

A. `-(2GMm)/(r)`
B. `-(GMm)/(r)`
C. `(GMm)/(2r)`
D. zero

Explanation

To solve the problem of two satellites colliding inelastically and determining the total energy of the system just after the collision, we can follow these steps:

### Step 1: Understand the System
We have two satellites of the same mass \( m \) orbiting the Earth at a radius \( r \). They are moving in opposite directions.

### Step 2: Calculate the Potential Energy
The gravitational potential energy \( U \) of a satellite of mass \( m \) at a distance \( r \) from the center of the Earth (mass \( M \)) is given by the formula:
\[ U = -\frac{G M m}{r} \]
where \( G \) is the gravitational constant.

Since there are two satellites, the total potential energy \( U_{\text{total}} \) of the system is:
\[ U_{\text{total}} = 2 \times U = 2 \left(-\frac{G M m}{r}\right) = -\frac{2 G M m}{r} \]

### Step 3: Calculate the Kinetic Energy
Each satellite in orbit has kinetic energy \( K \) given by:
\[ K = \frac{1}{2} m v^2 \]
The orbital speed \( v \) of a satellite in circular orbit is given by:
\[ v = \sqrt{\frac{G M}{r}} \]
Thus, the kinetic energy for one satellite becomes:
\[ K = \frac{1}{2} m \left(\sqrt{\frac{G M}{r}}\right)^2 = \frac{1}{2} m \frac{G M}{r} \]
For two satellites, the total kinetic energy \( K_{\text{total}} \) is:
\[ K_{\text{total}} = 2K = 2 \left(\frac{1}{2} m \frac{G M}{r}\right) = \frac{m G M}{r} \]

### Step 4: Total Energy Before Collision
The total mechanical energy \( E \) of the system before the collision is the sum of the total kinetic energy and total potential energy:
\[ E_{\text{before}} = K_{\text{total}} + U_{\text{total}} = \frac{m G M}{r} - \frac{2 G M m}{r} = -\frac{G M m}{r} \]

### Step 5: Energy After Collision
After the inelastic collision, the two satellites stick together. The wreckage will have zero kinetic energy because they are now at rest relative to each other (as they collide inelastically). Therefore, the total energy of the system just after the collision will be equal to the potential energy of the wreckage:
\[ E_{\text{after}} = U_{\text{total}} = -\frac{2 G M m}{r} \]

### Final Answer
Thus, the total energy of the system just after the collision is:
\[ \boxed{-\frac{2 G M m}{r}} \]