If force times distance = kinetic energy, than why is it defined as 1/2 of that?

Answer 1

Force times distance is not kinetic energy. Force times distance is work. Work can appear in different forms. If you expend 100 Newton-meters of work pushing a dog very slowly up a frictionless inclined plane, the dog will have 100 Newton-meters of potential energy when you are done, and no kinetic energy, for it will stop moving when you stop pushing.

On the other hand, if you expend 100 Newton-meters of work pushing the dog along a horizontal frictionless slippy-slide, the dog will continue sliding after you stop. It will have no potential energy and 100 Newton-meters of kinetic energy.

Do you want to know how fast the dog will be going when you stop pushing? Use

E = (1/2) * m * v^2

Answer 2

F*X is work; if invested in kinetic energy, KE=F*X. You may be referring to the potential energy formula for a deflected spring, PE=0.5*F*X (also expressed as 0.5*K*X^2 or 0.5*F^2/K, where K is the spring constant). Since PE is the integral of force taken over X, and F as a function of X is not constant but proportional to X, PE = 0.5 * F (the force at the deflected position) * X.

Answer 3

The energy formula E = mv^2/2 is the integral of the momentum equation p = mv. Energy is the sum of all the instantaneous momenta for a given velocity and mass.