For any motion with constant acceleration over a time interval, the product of acceleration times distance travelled over that interval is equal is equal to half the change in the square of the speed in the direction of acceleration. (This is just the junior high school version of the calculus identity \int{x’’ dx}=\int{x’’ x’ dt}=\int{x’ x’’ dt}=\int{x’ dx’}=\Delta(x’^2/2).)

So the quantity that is increased by applying force through a distance (to do “work” on a particle) is quadratic in its speed. But why is this important enough to give it a special name? That is because it allows us to define a quantity that is conserved throughout the evolution of any physical system.

As a result of Newton’s law of action and reaction (which is basically just a way of expressing conservation of momentum), in the motion of any system of particles the sum of mv^2/2 for all the particles plus the net work done against outside forces is a constant. We call this the “Energy” of the system and identify the part involving the speeds (that does not include work against the outside world) as the “kinetic” part of that energy – and the outside work (which includes an arbitrary constant depending on what we take as the starting point) is a called “potential” energy since it could in principle be returned to the system in future interactions.