Work–Energy Theorem

Whatever the forces, the net work done on an object shows up entirely as a change in its kinetic energy: W = ½m(v² − v₀²). Accelerate a 2 kg mass from rest to 10 m/s and exactly 100 J went in, no matter whether it took 1 m of huge force or 100 m of gentle push. Gaspard-Gustave de Coriolis formalised both "work" and the ½mv² form of kinetic energy in 1829, precisely so factory owners could compare what different machines actually delivered.

The theorem's power is that it skips time entirely, making it the fastest route to braking distances: a 1360 kg car slowing from 26.8 m/s to 13.4 m/s sheds about 367 kJ, and dividing that by the braking force gives the stopping distance directly. Sign discipline is the trap — friction and braking do negative work, so W comes out negative whenever the object slows, and this calculator will happily return a negative number to tell you so.