Centre of Mass>Velocity of Centre of Mass

Imagine a system of multiple objects moving in different directions with different speeds—like balls scattered on a moving train. While each ball has its own velocity, the entire system (train + balls) moves together in a specific way. The velocity of the centre of mass describes this overall motion of the system as a whole.​

Key Idea: The centre of mass velocity tells us how the "average position" of all the mass in a system is moving, regardless of how individual particles are moving.​

Consider a system of n particles with:

• Masses: m₁,m₂,m₃,…,m_n

• Individual velocities: \[\vec v_1\], \[\vec v_2\], \[\vec v_3\],..., \[\vec v_n\]

The velocity of the centre of mass (\[\vec v_{cm}\]) is defined as:​

\[\vec{v}_{CD}=\frac{m_1\vec{v}_1+m_2\vec{v}_2+m_3\vec{v}_3+\cdots+m_n\vec{v}_n}{m_1+m_2+m_3+\cdots+m_n}\]

This can be written more compactly as:

\[\vec{v}_{cm}=\frac{\sum_{i=1}^nm_i\vec{v}_i}{\sum_{i=1}^nm_i}=\frac{\sum_{i=1}^nm_i\vec{v}_i}{M}\]

where M is the total mass of the system.​

Remember: The centre of mass velocity is the weighted average of individual velocities, where the "weight" is the mass of each particle

When dealing with continuous objects (like rods, discs, or irregular shapes), we cannot count individual particles. Instead, we use integration.​​

For a continuous body:

\[\vec{v}_{cm}=\frac{1}{M}\int\vec{v}dm\]

where:

• dmdm is an infinitesimally small mass element
•  is the velocity of that mass element
• M is the total mass of the body
• The integration is performed over the entire body​

Physical Meaning: We imagine dividing the continuous body into countless tiny particles, finding the velocity contribution of each, and adding them all up (integrating).

Example 1: Exploding Projectile​

A projectile is fired and explodes mid-air into fragments. Even though the fragments scatter in all directions, the centre of mass continues along the original parabolic path as if no explosion occurred! This is because internal explosion forces don't change the system's total momentum.

Example 2: Rocket Propulsion​

As a rocket ejects fuel downward, the fuel and rocket move in opposite directions. The centre of mass of the (rocket + fuel) system moves smoothly, even though the rocket accelerates upward.

Example 3: Binary Star Systems​

Two stars orbiting each other follow complex paths. However, their centre of mass moves in a straight line with constant velocity (if no external forces act), making astronomical calculations much simpler.

Example 4: Collision Analysis​

In car crashes or ball collisions, tracking the centre of mass velocity helps predict the outcome, as the total momentum is conserved (when external forces are negligible).