Kinetic Energy

Kinetic energy is the energy an object carries because it is moving, $E_k = \tfrac{1}{2}mv^2$, and equivalently it is the work you would have to do to bring it from rest up to that speed — or the work it can do on something else in coming back to rest. The two odd-looking features, the half and the square, both fall out of that second statement. Push with force $F = ma$ through a distance and integrate: $\int ma\,\mathrm{d}x = \int mv\,\mathrm{d}v = \tfrac{1}{2}mv^2$. The square is not a modelling choice, it is what the integral hands back.

A 1500 kg car at 50 km/h — 13.9 m/s — carries $\tfrac{1}{2} \times 1500 \times 13.9^2 \approx 145$ kJ. The same car at 100 km/h, 27.8 m/s, carries about 580 kJ. Twice the speed, four times the energy, and since the brakes can only dissipate energy at roughly a fixed rate per metre of road, roughly four times the distance to stop.

Which form of energy is the "real" one was a genuine dispute. Descartes and his followers backed $mv$; Leibniz argued for what he called vis viva, $mv^2$. Willem 's Gravesande settled the experimental half of it in the 1720s by dropping brass balls into soft clay and finding that a ball arriving twice as fast sank about four times as deep, and Émilie du Châtelet made the theoretical case in the 1740s alongside her translation and commentary on the Principia. Both quantities turned out to matter — momentum $mv$ is conserved in every collision, kinetic energy only in elastic ones — which is why this site has pages for each.

Everything that goes wrong here goes wrong at the square. Doubling the speed does not double the energy, and the intuition that it does is what makes highway speeds feel deceptively similar to city ones. The unit trap follows directly: enter a speed in km/h where the formula wants m/s and you are wrong by $3.6^2 = 12.96$, not by 3.6 — the error is an order of magnitude and it looks plausible. Two smaller ones. Kinetic energy is a scalar with no direction, so two cars closing head-on do not have "negative" energy relative to each other, and you cannot cancel them the way you cancel momenta. And it is frame-dependent: a coffee cup on a train table has zero kinetic energy in your frame and a great deal in the frame of the platform. That is not a flaw; it is why the work–energy theorem only ever deals in changes.