The Work-Energy Theorem

Energy bookkeeping works because work is defined exactly so it matches kinetic energy. When a net force acts on an object over some displacement, it speeds the object up or slows it down, and the work-energy theorem says that change in motion is precisely W_net = KE_f - KE_i = (1/2)mv_f² - (1/2)mv_i^2. No other term sneaks in, because W_net already accounts for every force acting on the object, friction, gravity, applied pushes, all of it combined into one number.

You can derive this from a constant net force F acting over a distance d. Work is W = Fd. Newton's second law gives F = ma, so W = mad. The kinematic equation v_f² = v_i² + 2ad rearranges to ad = (v_f² - v_i²)/2. Substitute that in: W = m(v_f² - v_i²)/2 = (1/2)mv_f² - (1/2)mv_i^2. That is exactly the change in kinetic energy, so the theorem falls straight out of F = ma and one kinematic equation, no new physics required, just a repackaging.

This theorem is the simplest energy tool in the unit: it only tracks kinetic energy, with no potential energy or bar charts involved yet. Once potential energy enters, later in the unit, you will split W_net into work done by conservative forces, folded into PE, and work done by everything else. The work-energy theorem is the seed that grows into full energy conservation.