A ball kept in a close box moves in the box making collisions with the walls. The box is kept on a smooth surface. The velocity of the centre of mass

Options

A. of the box remains constant
B. of the box plus the ball sytem remains constant
C. of the ball remains constant
D. of the ball relative to the box remains constant.

Answer

To solve the problem regarding the motion of a ball inside a closed box on a smooth surface, we will analyze the situation step by step.

### Step-by-Step Solution:

1. **Understanding the System**:
- We have a ball inside a closed box. The box is placed on a smooth surface, which means there is negligible friction acting on the box.

2. **Identifying Forces**:
- When the ball collides with the walls of the box, it exerts a force on the wall, and the wall exerts an equal and opposite force on the ball (Newton's Third Law). This interaction is an internal force within the system (ball + box).

3. **Momentum Conservation**:
- Since the forces acting during the collision are internal to the system (ball and box), the total momentum of the system (ball + box) is conserved. There are no external forces acting on the system.

4. **Center of Mass (COM) Concept**:
- The center of mass of a system is defined as the point where the total mass of the system can be considered to be concentrated. The velocity of the center of mass (V_COM) is given by the total momentum of the system divided by the total mass.

5. **Acceleration of the Center of Mass**:
- Since there are no external forces acting on the system, the acceleration of the center of mass (a_COM) is zero. This implies that the velocity of the center of mass remains constant.

6. **Conclusion**:
- Therefore, the velocity of the center of mass of the system (ball + box) does not change as the ball moves and collides with the walls of the box. It remains constant throughout the motion.

### Final Answer:
The velocity of the center of mass of the system (ball + box) remains constant.