Conservation of Energy

Kinetic, potential, and the work-energy theorem.

Work-energy theorem

The net work done on a particle equals its change in kinetic energy:

W_{\text{net}} = \Delta K = \tfrac{1}{2}mv_f^2 - \tfrac{1}{2}mv_i^2

For conservative forces, total mechanical energy is constant:

E = K + U = \text{const}

Dropping a mass $m$ from height $h$ , energy conservation gives the impact speed directly:

mgh = \tfrac{1}{2}mv^2 \implies v = \sqrt{2gh}

No need to solve equations of motion — that is the power of conservation laws.