Two particles A and B, initially at rest, moves towards each other under a mutual force of attraction. At the instant when the speed of A is v and the speed of B is 2 v, the speed of centre of mass is

Options

A. zero
B. v
C. 1.5 v
D. 3 v

Explanation

To find the speed of the center of mass of the two particles A and B, we can follow these steps:

### Step 1: Understand the system
Initially, both particles A and B are at rest. This means that their initial velocities are:
- \( u_A = 0 \) (initial velocity of A)
- \( u_B = 0 \) (initial velocity of B)

### Step 2: Analyze the final velocities
At a certain moment, the speeds of the particles are given as:
- \( v_A = v \) (final velocity of A)
- \( v_B = 2v \) (final velocity of B)

### Step 3: Apply the principle of conservation of momentum
Since there are no external forces acting on the system, the total momentum before and after must be conserved. The initial momentum \( P_i \) is given by:
\[ P_i = m_A \cdot u_A + m_B \cdot u_B = m_A \cdot 0 + m_B \cdot 0 = 0 \]

The final momentum \( P_f \) is given by:
\[ P_f = m_A \cdot v_A + m_B \cdot v_B = m_A \cdot v + m_B \cdot (2v) \]

### Step 4: Set initial momentum equal to final momentum
Since the initial momentum is zero, we have:
\[ 0 = m_A \cdot v + m_B \cdot (2v) \]

### Step 5: Rearranging the equation
This can be rearranged to:
\[ m_A \cdot v + 2m_B \cdot v = 0 \]
Factoring out \( v \):
\[ v(m_A + 2m_B) = 0 \]

### Step 6: Solve for the center of mass speed
Since \( v \) cannot be zero (as it is the speed of particle A), we conclude that:
\[ m_A + 2m_B = 0 \]
This implies that the system's center of mass must be stationary because the total momentum remains zero.

### Step 7: Conclusion
Thus, the speed of the center of mass \( V_{cm} \) is:
\[ V_{cm} = 0 \]

### Final Answer
The speed of the center of mass is \( 0 \).

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