Conservation of Momentum (or equivalently Newton’s Law of action and reaction) tells us that when two particles interact just with one another, the acceleration of each multiplied by its mass (which we refer to as “Forces”) are equal and opposite. For any motion, the accumulated product of acceleration during a time interval times distance travelled over that interval is equal is equal to half of the change in the square of the speed in the direction of acceleration.

(This is just the calculus identity [math]\int{x’’ dx}=\int{x’’ x’ dt}=\int{x’ x’’ dt}=\int{x’ dx’}=\Delta(x’^2/2)[/math]; but if that’s not familiar to you, then a more elementary version is the fact that for constant acceleration with [math]v=at[/math] at time [math]t[/math], the average speed from time [math]0[/math] to time [math]t[/math] is [math]\frac{1}{2}at[/math] , so the distance travelled is given by [math]x=(\frac{1}{2}at)t=at^2/2[/math] , and acceleration times distance is [math]ax=a(at^2/2)=(at)^2/2=v^2/2[/math].)

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 in which the Forces between particles depend on their relative displacement only(*), and have no other dependence on time. This ensures that the speed lost when moving against forces (such as when a projectile moving away from a planet slows down) can be recovered if the motion is later reversed. So in the motion of any such system of particles, the sum of [math]mv^2/2[/math] for all the particles plus the net work done against forces is a constant. We call this the “Energy” of the system and identify the part involving the speeds as the “kinetic” part of that energy – and the work done against forces (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.

(*)- If we allow velocity-dependent forces such as friction then the process might not be reversible and we might have to include also other kinds of energy such as heat in order to still have a conserved quantity.