Consider a system of two identical particles. One of the particle is at rest and other has an acceleration `vec(a)`. The centre of mass has an acceleration

Options

A. zero
B. `(1)/(2)a`
C. `vec(a)`
D. `2vec(a)`

Answer

To solve the problem of finding the acceleration of the center of mass of a system of two identical particles, we can follow these steps:

### Step 1: Define the system
We have two identical particles, each with mass \( m \). One particle is at rest, and the other particle has an acceleration \( \vec{a} \).

### Step 2: Identify the accelerations of the particles
Let:
- Particle 1 (at rest): \( \vec{a_1} = 0 \)
- Particle 2 (accelerating): \( \vec{a_2} = \vec{a} \)

### Step 3: Write the formula for the acceleration of the center of mass
The acceleration of the center of mass \( \vec{a_{cm}} \) for a system of particles is given by the formula:

\[ \vec{a_{cm}} = \frac{\sum m_i \vec{a_i}}{\sum m_i} \]

where \( m_i \) is the mass of the \( i \)-th particle and \( \vec{a_i} \) is its acceleration.

### Step 4: Substitute the values into the formula
In our case, we have:
- For Particle 1: \( m_1 = m \) and \( \vec{a_1} = 0 \)
- For Particle 2: \( m_2 = m \) and \( \vec{a_2} = \vec{a} \)

Substituting these values into the formula gives:

\[ \vec{a_{cm}} = \frac{m \cdot 0 + m \cdot \vec{a}}{m + m} \]

### Step 5: Simplify the expression
This simplifies to:

\[ \vec{a_{cm}} = \frac{0 + m \vec{a}}{2m} = \frac{m \vec{a}}{2m} = \frac{\vec{a}}{2} \]

### Step 6: Conclusion
Thus, the acceleration of the center of mass is:

\[ \vec{a_{cm}} = \frac{\vec{a}}{2} \]

### Final Answer
The acceleration of the center of mass is \( \frac{\vec{a}}{2} \).

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