A nonzero external force on a system of particles. The velocity and the acceleration of the cente of mass are found to be `v_0 and a_0` at an instant t. It is possible that

Options

A. `v_0=0, a_0=0`
B. `v_0=0, a_0!=0`
C. `v_0!=0,a_0=0`
D. `v_0!=0,a_0!=0`

Answer

To solve the problem, we need to analyze the relationship between the external force acting on a system of particles, the acceleration of the center of mass, and the velocity of the center of mass.

### Step-by-Step Solution:

1. **Understanding the System**: We have a system of particles subjected to a non-zero external force. The velocity of the center of mass is denoted as \( v_0 \) and its acceleration as \( a_0 \) at a certain instant \( t \).

2. **Relation between Force and Acceleration**: According to Newton's second law, the acceleration of the center of mass \( a_{cm} \) of a system of particles is given by:
\[ a_{cm} = \frac{F_{external}}{M} \]
where \( F_{external} \) is the net external force acting on the system and \( M \) is the total mass of the system.

3. **Given Conditions**: We know that:
- \( F_{external} \neq 0 \) (since a non-zero external force is acting).
- Therefore, \( a_{cm} \) must also be non-zero because the total mass \( M \) is a positive quantity.

4. **Acceleration of the Center of Mass**: Since \( a_0 \) is the acceleration of the center of mass at time \( t \), we can conclude that:
\[ a_0 \neq 0 \]
This indicates that the center of mass is accelerating due to the external force.

5. **Velocity of the Center of Mass**: The velocity \( v_0 \) of the center of mass can be either positive, negative, or zero. However, the fact that there is a non-zero acceleration \( a_0 \) means that the velocity can be changing. Thus, \( v_0 \) can take any value (positive, negative, or zero).

6. **Conclusion**: It is possible that:
- The center of mass has a non-zero velocity \( v_0 \) and a non-zero acceleration \( a_0 \).
- The center of mass could also have zero velocity \( v_0 = 0 \) while still having a non-zero acceleration \( a_0 \).

### Final Answer:
It is possible that \( v_0 \) can be any value (positive, negative, or zero), while \( a_0 \) must be non-zero due to the presence of a non-zero external force.

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