Two bodies of masses `m_1` and `m_2` are moving with velocity `v_1` and `v_2` respectively in the same direction. The total momentum of the system in the frame of reference attached to the centre of mass is (`v` is relative velocity between the masses)

Options

A. `(m_(1)m_(2)v)/(m_(1)-m_(2))`
B. `(2m_(1)m_(2)v)/(m_(1)+m_(2))`
C. zero
D. `(4m_(1)m_(2)v)/(m_(1)+m_(2))`

Answer

To solve the problem of finding the total momentum of a system of two bodies in the center of mass frame, we can follow these steps:

### Step 1: Define the System
We have two bodies with masses \( m_1 \) and \( m_2 \) moving with velocities \( v_1 \) and \( v_2 \) respectively in the same direction.

### Step 2: Find the Velocity of the Center of Mass (v_cm)
The velocity of the center of mass \( v_{cm} \) of the system can be calculated using the formula:
\[ v_{cm} = \frac{m_1 v_1 + m_2 v_2}{m_1 + m_2} \]

### Step 3: Determine the Velocities in the Center of Mass Frame
In the center of mass frame, the velocities of the two bodies relative to the center of mass are:
- For mass \( m_1 \):
\[ v_{1, cm} = v_1 - v_{cm} \]
- For mass \( m_2 \):
\[ v_{2, cm} = v_2 - v_{cm} \]

### Step 4: Calculate the Momentum in the Center of Mass Frame
The momentum of each body in the center of mass frame is given by:
- Momentum of \( m_1 \):
\[ p_1 = m_1 (v_1 - v_{cm}) \]
- Momentum of \( m_2 \):
\[ p_2 = m_2 (v_2 - v_{cm}) \]

### Step 5: Total Momentum in the Center of Mass Frame
The total momentum \( P \) of the system in the center of mass frame is:
\[ P = p_1 + p_2 = m_1 (v_1 - v_{cm}) + m_2 (v_2 - v_{cm}) \]

### Step 6: Substitute the Expression for \( v_{cm} \)
Substituting \( v_{cm} \) into the equation:
\[ P = m_1 (v_1 - \frac{m_1 v_1 + m_2 v_2}{m_1 + m_2}) + m_2 (v_2 - \frac{m_1 v_1 + m_2 v_2}{m_1 + m_2}) \]

### Step 7: Simplify the Expression
Combining the terms:
\[ P = m_1 v_1 - \frac{m_1 (m_1 v_1 + m_2 v_2)}{m_1 + m_2} + m_2 v_2 - \frac{m_2 (m_1 v_1 + m_2 v_2)}{m_1 + m_2} \]
This simplifies to:
\[ P = \frac{m_1 v_1 + m_2 v_2 - (m_1 v_1 + m_2 v_2)}{m_1 + m_2} = 0 \]

### Conclusion
Thus, the total momentum of the system in the center of mass frame is:
\[ \text{Total Momentum} = 0 \]