Proof of the Law of Conservation of Momentum
It is not obvious why momentum should be conserved. Momentum is avery abstract concept, and the principle of conservation of momentummay be seen as a consequence of other, fundamental Laws of physics:
The principle of Relativity
The Principle of Conservation of Energy
In the inertial frame O' above the particle is moving withvelocityinthe y' direction, and it's kinetic energy isandif the particle moves in a potential field V due to particles atpositions relative toofithas potential energyInthe inertial frame O', the velocity of the particle isandit's kinetic energy isandthe Galilean transformation givessothat
The sum of all the energies in each inertial frame by the Law ofConservation of energy.
Hence the difference of the two energies is constant.
isconstant and so isthereforeisa constant vector.