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Momentum Conservation

The principle of conservation of momentum states that the momentum p = m v of a (point) particle, of mass m moving with linear velocity v, remains constant in time, as observed from an inertial frame, if the particle is not subjected to any net external force. This follows as the first integral of the Newton’s II law (i.e. equation of motion) for the isolated particle. If we take two isolated particles interacting with each other, the nature of forces of interaction between them (Newton’s III law) leads to conservation of total momentum (P = p₁ + p₂) for the combined system of two particles. Conservation of momentum is needed a consequence of the concept of homogeneity of space for isolated system of interacting particles.

Thus, in absence of external force, the momentum of a particle of a system of particles remain constant in time. However, we never precisely defined the momentum of a ‘system of particles’. Newton’s laws of motion, were articulated only for a (point) particle: the position vector r defines a point and hence v = dr/dt has meaning only for a (point) particle. We, of course, used the words like object, body, system, etc. indiscriminately when we meant only a particle.

We shall therefore, define the momentum of an extended object (i.e. a system of particles) and introduce the concept of centre of mass of the object. Newton’s law and momentum, conservation principle will ne reformulated for such an extended object. Finally, we shall discuss the application of momentum conservation to understand two specific processes: the phenomenon of collision of two (and more) bodies, and the dynamics of an object with variable mass (the rocket problem).