For the more mathematically minded..
For those of you with a mathematical bent, here is a brief review of (1D) potential and kinetic energy from the Newtonian viewpoint.
Kinetic energy and the Work-Energy Theorem
Let’s start with Newton’s second law (Key Point 2.2):
F=maUsing the chain rule we rewrite the acceleration:
If we differentiate v2 w.r.t x, we get:
Therefore we can rewrite Newton’s second law as
where K is the kinetic energy.
Integrating both sides, we obtain the work-energy theorem (Key Point 3.7) in one dimension:
Potential Energy
Now let’s assume we can define a function U(x) such that
Inserting this into the work energy theorem
Rearranging, we get that (Key Point 3.16)
Ki+Ui=Kf+UfThis is the law of conservation of energy: total mechanical energy is conserved.
- Naturally this only applies if we can write F=-dU/dx. It is possible to show that this is a necessary and sufficient condition for the force to be conservative.