My understanding of energy is that it is a quantity assigned to a system that is, in some way, related to its physical properties. This quantity remains constant when a system changes its state unless there is an interaction between two systems in a specific way.
This quantity can take various forms, one of which is called Kinetic Energy.
We define it as:
$K = \frac{1}{2}mv^2$
The best reason I’ve found for defining it this way is that when we relate the work done on a body with its motion (or velocity), we get the following relation, better known as the work-energy theorem:
$W_{AB} = \frac{1}{2}m{v_A}^2 - \frac{1}{2}m{v_B}^2$
Defining kinetic energy in this way allows us to write this relation more nicely, and perhaps more aesthetically, as:
$W_{AB} = \Delta K_{AB}$
1. Why did we have to include this $\frac{1}{2}mv^2$ term as part of our "magic number" of energy?
If we just wanted a nice trick, couldn’t we have called this $\frac{1}{2}mv^2$ term something else—like the kinectivity of the object—and still obtained the same relation between work and kinectivity? Why did we need to relate it to our "magic number" (energy)?
Relating $\frac{1}{2}mv^2$ to energy introduces a complication: the total energy of a system can now change without the system interacting with another system in the specified way by just application of force. To fix this, we introduce the concept of Potential Energy. Potential energy is essentially a trick to balance changes in kinetic energy and ensure that the net change in energy of the system (more precisely, the mechanical energy) remains zero.
I cannot fully grasp why we came up with the concept of energy in the first place, or why we introduced the idea of work (though I can somewhat see it as a way to quantify the influence of force on a body’s displacement). Surely, it isn’t all just a trick to make our calculations easier.
Studying motion seems natural. The question of why motion happens leads us naturally to forces, but energy feels synthetic.
If we hadn’t called $\frac{1}{2}mv^2$ a component of energy, then energy wouldn’t have the unit of joules, and we wouldn’t have the concept of potential energy either.
I am not interested in a historical explanation but rather the thought process that led to the definition of kinetic energy and energy in general.
Kinetic Energy equation: Is $K=\frac12mv^2$ a Definition, or a derived Theorem?
This question has answer which are somewhat more historical, and they do not clear my doubts raised above.
I hope I have been able to explain my concerns clearly. Any response is highly appreciated.
