Two particles of equal mass have velocities `v1=2 i ms^(-1)` and `v2= 2 j ms^(-1)` First particle has an acceleration `a= (3i + 3j) ms^(-2)` while the acceleration of the other particle is zero. The centre of mass of the two particles moves in a path of

Options

A. a.straight line
B. b.parabola
C. c.circle
D. d.ellipse

Explanation

To solve the problem, we need to determine the path of the center of mass of two particles with given velocities and accelerations. Here’s a step-by-step solution:

### Step 1: Identify the given data
We have two particles with:
- Mass of each particle: \( m \)
- Velocity of particle 1: \( \mathbf{v_1} = 2 \mathbf{i} \, \text{m/s} \)
- Velocity of particle 2: \( \mathbf{v_2} = 2 \mathbf{j} \, \text{m/s} \)
- Acceleration of particle 1: \( \mathbf{a_1} = 3 \mathbf{i} + 3 \mathbf{j} \, \text{m/s}^2 \)
- Acceleration of particle 2: \( \mathbf{a_2} = 0 \, \text{m/s}^2 \)

### Step 2: Calculate the velocity of the center of mass (\( \mathbf{v_{cm}} \))
The formula for the velocity of the center of mass is given by:
\[ \mathbf{v_{cm}} = \frac{m_1 \mathbf{v_1} + m_2 \mathbf{v_2}}{m_1 + m_2} \]
Since both particles have equal mass \( m \), we can simplify:
\[ \mathbf{v_{cm}} = \frac{m \mathbf{v_1} + m \mathbf{v_2}}{m + m} = \frac{\mathbf{v_1} + \mathbf{v_2}}{2} \]
Substituting the values:
\[ \mathbf{v_{cm}} = \frac{2 \mathbf{i} + 2 \mathbf{j}}{2} = \mathbf{i} + \mathbf{j} \, \text{m/s} \]

### Step 3: Calculate the acceleration of the center of mass (\( \mathbf{a_{cm}} \))
The formula for the acceleration of the center of mass is:
\[ \mathbf{a_{cm}} = \frac{m_1 \mathbf{a_1} + m_2 \mathbf{a_2}}{m_1 + m_2} \]
Again, since both particles have equal mass \( m \):
\[ \mathbf{a_{cm}} = \frac{m \mathbf{a_1} + m \mathbf{a_2}}{m + m} = \frac{\mathbf{a_1} + \mathbf{a_2}}{2} \]
Substituting the values:
\[ \mathbf{a_{cm}} = \frac{(3 \mathbf{i} + 3 \mathbf{j}) + 0}{2} = \frac{3 \mathbf{i} + 3 \mathbf{j}}{2} = \frac{3}{2} \mathbf{i} + \frac{3}{2} \mathbf{j} \, \text{m/s}^2 \]

### Step 4: Analyze the relationship between \( \mathbf{v_{cm}} \) and \( \mathbf{a_{cm}} \)
Both \( \mathbf{v_{cm}} \) and \( \mathbf{a_{cm}} \) are in the same direction:
- \( \mathbf{v_{cm}} = \mathbf{i} + \mathbf{j} \)
- \( \mathbf{a_{cm}} = \frac{3}{2} \mathbf{i} + \frac{3}{2} \mathbf{j} \)

Since both vectors are parallel, this indicates that the center of mass moves in a straight line.

### Conclusion
The center of mass of the two particles moves in a straight line.