Two particles of different masses are thrown simultaneously in air. Centre of mass of two balls while both are in mid air.

Options

A. Move in a straight line independent of initial velocities of particles
B. Move in straight line when both the particles are thrown along same vertical line
C. Have acceleration equal to g
D. Have acceleration depending on initial velocities of particles

Explanation

To solve the problem of finding the behavior of the center of mass of two particles of different masses thrown simultaneously in the air, we can analyze the situation step by step.

### Step-by-Step Solution:

1. **Understanding the System**:
We have two particles with different masses, \( m_1 \) and \( m_2 \). They are thrown simultaneously into the air. We need to determine the motion of their center of mass.

2. **Defining the Center of Mass**:
The center of mass (CM) of a system of particles is given by the formula:
\[ \text{CM} = \frac{m_1 \cdot x_1 + m_2 \cdot x_2}{m_1 + m_2} \]
where \( x_1 \) and \( x_2 \) are the positions of the two particles.

3. **Analyzing the Motion**:
- If both particles are thrown at the same angle and with different velocities, they will follow parabolic trajectories due to the influence of gravity.
- The center of mass will also follow a parabolic path, but its trajectory will depend on the relative masses and initial velocities of the two particles.

4. **Special Case - Vertical Throw**:
- If both particles are thrown vertically (along the same vertical line), the center of mass will move in a straight line. This is because both particles are subjected to the same gravitational force and will have the same vertical motion.

5. **Acceleration of the Center of Mass**:
- The only external force acting on the system is the gravitational force. This means that the acceleration of the center of mass will be equal to \( g \) (acceleration due to gravity), regardless of the initial velocities of the particles.

6. **Conclusion**:
- The center of mass of the two particles moves in a straight line when both are thrown vertically.
- The center of mass has an acceleration equal to \( g \), independent of the initial velocities of the particles.

### Final Answer:
- The correct statements regarding the center of mass of the two particles while both are in mid-air are:
- It moves in a straight line when both particles are thrown along the same vertical line.
- It has an acceleration equal to \( g \).