Two satellites of same mass m are revolving round of earth (mass M) in the same orbit of radius r. Rotational directions of the two are opposite therefore, they can collide. Total mechanical energy of the system (both satallites and earths) is `(m lt lt M)` :-

Options

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

Answer

To solve the problem, we need to calculate the total mechanical energy of the system, which includes two satellites and the Earth. We will follow these steps:

### Step 1: Understand the System
We have two satellites, each with mass \( m \), revolving around the Earth with mass \( M \) at a distance \( r \) from the center of the Earth. The satellites are moving in opposite directions.

### Step 2: Calculate the Gravitational Potential Energy (U)
The gravitational potential energy \( U \) of one satellite in the gravitational field of the Earth is given by the formula:
\[ U = -\frac{G M m}{r} \]
where \( G \) is the gravitational constant.

### Step 3: Calculate the Kinetic Energy (K)
The kinetic energy \( K \) of one satellite can be derived from the centripetal force balance. The gravitational force provides the necessary centripetal force for the satellite's circular motion:
\[ \frac{mv^2}{r} = \frac{G M m}{r^2} \]
From this, we can solve for \( v^2 \):
\[ mv^2 = \frac{G M m}{r} \implies v^2 = \frac{G M}{r} \]
Now, substituting \( v^2 \) into the kinetic energy formula:
\[ K = \frac{1}{2} mv^2 = \frac{1}{2} m \left(\frac{G M}{r}\right) = \frac{G M m}{2r} \]

### Step 4: Calculate the Total Mechanical Energy (E) for One Satellite
The total mechanical energy \( E \) of one satellite is the sum of its potential and kinetic energy:
\[ E = U + K = -\frac{G M m}{r} + \frac{G M m}{2r} \]
Combining these:
\[ E = -\frac{G M m}{r} + \frac{G M m}{2r} = -\frac{G M m}{2r} \]

### Step 5: Calculate the Total Mechanical Energy for Both Satellites
Since both satellites have the same mass and are in the same orbit, the total mechanical energy of the system (both satellites) is:
\[ E_{\text{total}} = E_A + E_B = -\frac{G M m}{2r} + -\frac{G M m}{2r} = -\frac{G M m}{r} \]

### Final Answer
Thus, the total mechanical energy of the system (both satellites and Earth) is:
\[ \boxed{-\frac{G M m}{r}} \]