How Acceleration Relates to Kinetic Energy

How a mass accelerates is an interesting question.  When a force is applied to an object, that object's momentum changes as well as (sometimes, as explained below) its kinetic energy.  For this question, I'll focus mainly on how a force affects kinetic energy.

At low speeds and energies, all of the forces acting on an object equal that object's mass times its acceleration (called Newton's 2nd law).  The relationship between force and energy can be derived from the aforementioned 2nd law:

So we have first F=ma (Newton's 2nd law) where F is force, m is mass, and a is acceleration.  It should be noted that any bold letters are vectors meaning that they have magnitude and direction.

This can be rewritten more generally as F=(dp/dt) where p is momentum and dp/dt implies a change in momentum with respect to a change in time.  Momentum, p, however, is related to kinetic energy, KE, by the equation KE=p²/2m.  So a change in momentum corresponds to a change in kinetic energy.

This is the essence of Newton's second law: Applying a force to a mass changes the  momentum of that mass.  An acceleration just represents this change in momentum for an object that has a constant mass.

The units newtons and joules can be connected directly.  For a mass under a constant force, F^.Δx=ΔKE where the Δ is the symbol for change and the "^." means that only the part of F that is in the same direction as x should be multiplied.  So moving an object a distance x with a force F changes the kinetic energy in a mathematically direct fashion.

This description of forces, masses, and energies is a little simplistic.  and have more information on derivations of kinetic energy and how it relates to force.  Wikipedia's article has some mathematical rigor but both are very informative.