Two particles of equal mass have velocities `vec v_1 = 2 hat i= m//s^-1` and `vec v_2 = 2hat j m//s^-1`. First particle has an acceleration `vec a_1= (3 hat i+ 3 hat j) 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. straight line
B. parabola
C. circle
D. ellipse

Explanation

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

### Step 1: Identify the velocities of the particles
The velocities of the two particles are given as:
- Particle 1: \(\vec{v_1} = 2 \hat{i} \, \text{m/s}\)
- Particle 2: \(\vec{v_2} = 2 \hat{j} \, \text{m/s}\)

### Step 2: Calculate the velocity of the center of mass
The formula for the velocity of the center of mass (\(\vec{v_{cm}}\)) of two particles of equal mass is:
\[ \vec{v_{cm}} = \frac{m \vec{v_1} + m \vec{v_2}}{m + m} = \frac{\vec{v_1} + \vec{v_2}}{2} \]
Substituting the values:
\[ \vec{v_{cm}} = \frac{2 \hat{i} + 2 \hat{j}}{2} = \hat{i} + \hat{j} \, \text{m/s} \]

### Step 3: Identify the accelerations of the particles
The accelerations of the particles are:
- Particle 1: \(\vec{a_1} = 3 \hat{i} + 3 \hat{j} \, \text{m/s}^2\)
- Particle 2: \(\vec{a_2} = 0 \, \text{m/s}^2\)

### Step 4: Calculate the acceleration of the center of mass
The formula for the acceleration of the center of mass (\(\vec{a_{cm}}\)) is:
\[ \vec{a_{cm}} = \frac{m \vec{a_1} + m \vec{a_2}}{m + m} = \frac{\vec{a_1} + \vec{a_2}}{2} \]
Substituting the values:
\[ \vec{a_{cm}} = \frac{(3 \hat{i} + 3 \hat{j}) + 0}{2} = \frac{3 \hat{i} + 3 \hat{j}}{2} = \frac{3}{2} \hat{i} + \frac{3}{2} \hat{j} \, \text{m/s}^2 \]

### Step 5: Analyze the direction of velocity and acceleration
The velocity of the center of mass is \(\hat{i} + \hat{j}\), and the acceleration of the center of mass is \(\frac{3}{2} \hat{i} + \frac{3}{2} \hat{j}\). Both the velocity and acceleration vectors point in the same direction (the first quadrant).

### Step 6: Conclusion
Since the velocity and acceleration of the center of mass are in the same direction, the center of mass will move in a straight line.

Thus, the answer is:
**The center of mass of the two particles moves in a straight line.**