system
Given the following post, respond as if you're commenting on Reddit:
human
Is there any scientific reasoning behind the fact that momentum is the derivative of kinetic energy with respect to velocity?

Is this just a sort of mathematical coincidence or are the principles of one's momentum and kinetic energy intertwined?
gpt
It is definitely not a mathematical coincidence.  When a body is being accelerated its gaining kinetic energy according to:

dE=Fdx=vdp.  If we assume m is constant then we can do the following:

dE=vd(mv)=mvdv=pdv, and therefore dE/dv=p.  

Most modern physicists prefer to express KE as p^2  /2m because it makes the relationship between energy and momentum more explicit (and because this is how it occurs in the Hamiltonian formulation of classical mechanics.)