Question

It is been proved that Newton's third law implies the conservation of momentum for a system of particles if there is no net external force. For a System of interacting particles, prove the converse: if total momentum for the two particles is conserved in absence of a net external force, then the inter particle forces must obey third law of Newton.

Answer 1

Consider an isolated system of two bodies of masses mi and m2 mutually interacting with each other These interactions cause a change in the velocities and hence a change in the momentum of each body Let Δp1 and Δp2denote the change in momentum of m1 and m2 respectively brought about in a time interval Δ t. Now, according to the law of conservation of linear momentum, the net change in the linear momentum in an isolated system must be zero, i.e. Dividing by At and taking the limit as At A 0, we have dt dt i.e. rate of change of momentum of m2 =-rate of change of momentum ml or Force on m2 - Force of mi or Action- Reaction i.e. the action and reaction are always equal and opposite. This is Newton's third law of motion