Force, by definition, is the rate at which momentum changes with time. What is weight, but a force in the vertical direction.
Momentum = m*v
Force = Change in momentum = change in m/change in t * v + m*change in v/change in t
Since mass doesn't usually change, we typically ignore m/t*v, so the equation comes out to be...
Force = m * v/t
And change in velocity divided by the change in time is acceleration
F = m * a