It is not obvious why momentum should be conserved. Momentum is a very abstract concept, and the principle of conservation of momentum is really 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 with velocityin the y' direction, and it's kinetic energy isand if the particle experiences no external force this kinetic energy is constant. In the inertial frame O', the velocity of the particle isand it's kinetic energy is and since the particle is subject to no external forces in O', it is subject to no external forces in O, so this kinetic energy is constant.

Hence the difference of the two kinetic energies is constant.

is constant and so isthereforeis a constant vector and so isand we can now sum over many particles to obtain