Velocity of Centre of Mass:

The velocity of the centre of mass is defined as the rate at which the centre of mass of a system of particles changes its position with respect to time. It represents the overall motion of the system.

Velocity of Particle in Rolling Motion:

The velocity of a particle in rolling motion refers to the speed at which an individual particle on the rolling object moves. It represents the linear motion of the particle relative to the ground.

Difference between Velocity of Centre of Mass and Velocity of Particle in Rolling Motion:

1. Motion: The velocity of the centre of mass describes the motion of the entire system as a whole, while the velocity of a particle in rolling motion describes the motion of an individual particle on the rolling object.

2. Direction: The velocity of the centre of mass is always directed in the same direction as the overall motion of the system. On the other hand, the velocity of a particle in rolling motion can vary depending on the position of the particle on the rolling object. It can be in the same direction as the overall motion, opposite to it, or even zero.

3. Speed: The speed of the centre of mass is equal to the average linear speed of all the particles in the system. In the case of rolling motion, the speed of the particle can vary depending on its distance from the axis of rotation. The particles farther away from the axis of rotation have a higher linear speed compared to those closer to it.

4. Application: The velocity of the centre of mass is useful in analyzing the motion of objects and systems, especially in terms of conservation laws. On the other hand, the velocity of a particle in rolling motion is important in understanding the dynamics of rolling objects, such as the forces involved and the rolling resistance.

In conclusion, the velocity of the centre of mass and the velocity of a particle in rolling motion are related but represent different aspects of motion. The former describes the overall motion of the system, while the latter focuses on the motion of individual particles on the rolling object.