Forces

Constant Force

If f(x, t) is f u(t) (now the u(t) is dropped mostly in notation)…
then PE:
$P\left(x\right)=-\int fdx=-fx$
This can also be written in our conservation of mechanical energy notation
$\frac{m{v}^2}{2}-\frac{m{v}_0^2}{2}=f\left(x-x_0\right)$ Where changes in potential energy (R.S) becomes kinetic energy (L.S) since fx is P(x) which is P.E or work as we can recall.

Natural Language

By rearranging the equation, we can see that the final K.E of a system is its initial K.E + work done on the system.

Non Conservative Force of Dry Friction

The work done by friction is path dependent.

Friction

$F=\mu N$
${\mu }_s>{\mu }_d$

Friction dissipates as heat 🔥.