Why use momentum instead of velocity in classical mechanics

[SUDOnym]

In classical physics, the two quantities that you need to know about a particle to describe it completely at an instant is x and p - position and momentum.
Why is it that momentum is chosen as the second quantity - ie. couldn't we describe the particle equally well with x and v - position and velocity?
Am I missing something or is chosing momentum just convention?

 

[Matterwave]

Yes we can. In fact, in the Lagrangian formulation of Mechanics, it is x and v which evolve in time. For Hamiltonian mechanics, we choose x and p. The decision is not entirely convention, there are some insights into the nature of mechanics which are afforded to us by using the Hamiltonian formulation.

For more information, you can look up Hamiltonian Mechanics.