Energy Is Nature’s Accounting System

Energy gives physics a way to track change without narrating every instant of motion.

A quantity that survives change

For an isolated system, energy may change form while the total remains constant.

Why the method works

If only conservative forces act, then $K_i+U_i=K_f+U_f$.

Derive work–energy

Using $F=ma$ and $a=v\,dv/dx$,

$\int_{x_1}^{x_2}F\,dx=\int_{v_1}^{v_2}mv\,dv=\Delta\!\left(\frac12mv^2\right).$

Net work changes kinetic energy. For a conservative force, define $\Delta U=-W_c$ so that $K+U$ remains constant when no other transfer occurs.

Check your understanding

Can mechanical energy decrease while total energy is conserved?

Show the reasoning

Yes. Friction can convert kinetic or potential energy into internal thermal energy while total energy of a closed system remains constant.