Short answers, pulled from the story.
John Philoponus declared in the year 530 AD that a thrown stone carries an invisible force within itself called impetus. This idea challenged Aristotle's teaching that air pushes the stone forward. Philoponus argued that motion is sustained by a quality imparted to the object at the moment of release.
Isaac Newton published his Principia on the 5th of July 1687. He defined momentum as the quantity of motion arising from velocity and the quantity of matter conjointly. Newton's second law established that the change in this quantity is proportional to the force impressed.
In a perfectly inelastic collision, the two bodies move together after impact and 100 percent of the kinetic energy is converted into other forms of energy. The momentum remains conserved because the vector sum of the momenta before the collision equals the vector sum after. This principle holds true whether the collision is elastic or inelastic.
Rockets generate motion by thrusting propellant outward to gain an equal and opposite momentum through the conservation of momentum. The motion is entirely dependent on the momentum transfer between the vehicle and its exhaust. This principle applies in the vacuum of space where there is no air to push against.
The Heisenberg uncertainty principle defines the limits of how accurately the momentum and position of a single observable system can be known at once. It establishes that position and momentum are conjugate variables. For a single particle, the momentum operator involves the gradient operator and the reduced Planck constant.
Albert Einstein's special theory of relativity introduced the Lorentz transformation which relates position and time in two reference frames moving relative to one another. In this framework, the inertial mass of an object becomes a function of velocity and the modified momentum obeys Newton's second law when the relativistic factor is included. The four-momentum is invariant under Lorentz transformations implying the conservation of both mass and energy.