Past exam of the mathematical methods II course of the University of Oxford / 2014 / 12 / d

Parametrize any actual particle trajectory by time, with $\mathbf{v}=\dot{\mathbf{r}}$ and acceleration $\mathbf{F}=\dot{\mathbf{v}}$. The chain rule givesIntegrating from the origin to the endpoint givesHere $K$ is specific kinetic energy, because the source defines it without a mass factor. Equivalently, $|{\mathbf{v}}_A{|}^2-|{\mathbf{v}}_O{|}^2=2$ in the chosen units. The individual initial or final speeds are not fixed by this result. For a particle of mass $m$, the change of ordinary kinetic energy would be $mI$.