when momentum is linearly conserved the COM continues to have the same velocity right? so when we r talking about inelastic collision in 1-D,the relative velocity before & after the collision differ (right?) does this imply that the velocity of COM of the system has changed, coz if it does then C.O.L.M. IS NOT TRUE IN THIS CASE ....plz help.

in short, why is linear momentum conserved when 2 bodies collideand coefficient of restitution is between 0 and 1

You've raised an interesting point about momentum conservation, especially in the context of inelastic collisions. Let's break this down step by step to clarify how momentum works in these scenarios and why the center of mass (COM) velocity remains consistent even when the relative velocities of the colliding bodies change.

Understanding Momentum Conservation

In any closed system where no external forces act, the total linear momentum is conserved. This principle holds true regardless of the type of collision—elastic or inelastic. The key here is that while the individual velocities of the colliding bodies may change, the overall momentum of the system remains constant.

What Happens in an Inelastic Collision?

In an inelastic collision, two bodies collide and may stick together or deform, resulting in a loss of kinetic energy. However, the momentum before and after the collision is still conserved. Let's illustrate this with a simple example:

• Imagine two objects: Object A with mass m_A moving at velocity v_A and Object B with mass m_B at rest (velocity v_B = 0).
• The total momentum before the collision is given by: p_initial = m_A * v_A + m_B * v_B.
• After the collision, if they stick together, their combined mass is (m_A + m_B), and they move with a common velocity v_f.
• The total momentum after the collision is: p_final = (m_A + m_B) * v_f.

By the conservation of momentum, we set these equal: m_A * v_A = (m_A + m_B) * v_f. From this equation, you can solve for v_f, showing that the total momentum is conserved.

Relative Velocity and the Coefficient of Restitution

Now, regarding the relative velocities: in an inelastic collision, the coefficient of restitution (e) is between 0 and 1. This coefficient measures how elastic the collision is, with 0 indicating a perfectly inelastic collision (where the objects stick together) and 1 indicating a perfectly elastic collision (where they bounce off without losing kinetic energy).

In terms of relative velocity, before the collision, the relative velocity of approach is v_A - v_B. After the collision, the relative velocity of separation is given by e(v_A - v_B). Since e is less than 1, the relative velocity after the collision is less than before, indicating that the objects do not separate with the same speed they approached each other.

Center of Mass Velocity

Despite the changes in individual velocities, the velocity of the center of mass remains constant in the absence of external forces. The center of mass velocity (v_COM) can be calculated as:

Since the total momentum is conserved, the velocity of the center of mass does not change due to the collision itself. This is a crucial point: while the velocities of the individual bodies change, the overall motion of the center of mass remains unaffected by the internal interactions (the collision) as long as no external forces act on the system.

Final Thoughts

In summary, linear momentum is conserved during an inelastic collision because the total momentum before and after the collision remains constant, even though the individual velocities of the colliding bodies change. The center of mass continues to move with the same velocity, affirming the principle of conservation of momentum. Understanding these concepts helps clarify how momentum behaves in different types of collisions and reinforces the fundamental principles of physics.