What theories, laws, forumlas, equations, etc. can be used to prove and predict an object's motion?
At this level is convenient to classify "objects" as:
Particles; Rigid bodies; and other("loose")"systems of particles".
We will, for simplicity, assume the "object" to be a particle.
Newtons laws will give you the acceleration of a particle, GIVEN the mass of the particle and all the forces acting on the particle.
The forces acting on a particle depend on (1) the properties of our particle and (2) the objects "around" our particle,their properties, and their distances from our particle.
If these (1) and (2) are available,the force on our particle can be obtained from the set of "FORCE LAWS", partly listed below:
The Gravitational Force Law
The Electrostatic Force Law
The Friction Force Laws
The Tension Force Laws
Once you get the acceleration of a particle,knowing its INITIAL velocity, you can find the velocity at any subsequent instant of time