Here is the sentence that ties the chapter together: the net work done on a body equals the change in its kinetic energy. Not resembles. Equals. Written out, $W = \tfrac{1}{2} m \left(v^{2} - v_0^{2}\right)$, read aloud W equals one-half m, bracket, v squared minus v-nought squared. $W$ is the NET work in joules — everything pushing minus everything resisting; $m$ is the mass in kilograms; $v$ is the speed AFTER, and $v_0$ — said v-nought — is the speed BEFORE, both in metres per second. The subscript nought always means the beginning, in this room and everywhere else.

Order is not negotiable, and it carries the sign. Final minus initial: speed up and $W$ is positive, energy going in; slow down and $W$ is negative, energy coming out. A brake does negative work all day and is proud of it. If your answer for a braking section comes out positive, you have subtracted backwards — go and look, the arithmetic is fine.

Now the trap this theorem is famous for. $v^{2} - v_0^{2}$ is not $(v - v_0)^{2}$, and the two are not even close. Take 20 m/s down from 10: the difference of squares is 300, the square of the difference is 100. Say it as words and you will never do it again — difference of the squares, not square of the difference.

Why the theorem earns its keep: it gets you from a force and a distance to a SPEED without ever mentioning time or acceleration. And it is the honest way to see what a running start is worth — the sled that arrives already moving is charged only for the difference, and the faster it already goes, the more that difference costs.