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 withvelocity
inthe y' direction, and it's kinetic energy is
andif the particle moves in a potential field V due to particles atpositions relative to
of
ithas potential energy
Inthe inertial frame O', the velocity of the particle is
andit's kinetic energy is
andthe Galilean transformation 
gives
sothat
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 is
therefore
isa constant vector.