Kinetic Energy of a Mass

From Newton's second law, $f=ma=m{\ddot x}$ (introduced in Eq. (E.1)), we can derive the formula for the kinetic energy of a mass given its speed $v={\dot x}$. Let $d x$ denote a small (infinitesimal) displacement of the mass in the $x$ direction. Then we have, using the calculus of differentials,

$$\begin{eqnarray*} f(t) &=& m\, {\ddot x}\\ \,\,\implies\,\,\quad d W\isdef f(t)... ...c{1}{2}{\dot x}^2\right)\\ &=& d\left(\frac{1}{2}m\,v^2\right). \end{eqnarray*}$$

Thus, by Newton's second law, a differential of work $dW$ applied to a mass $m$ by force $f$ through distance $d x$ boosts the kinetic energy of the mass by $d(m\,v^2/2)=m\,v\,dv=ma\,dx$. Therefore, we must have

$$E_m(t) = \frac{1}{2}m v^2(t) = \frac{1}{2}{\dot x}^2(t),$$

where $E_m(t)$ denotes the kinetic energy of the mass $m$ traveling at speed $v(t)$ at time $t$.

The quantity $dW=f\,dx$ is classically called the virtual work associated with force $f$, and $d x$ a virtual displacement [521].