Mass Kinetic Energy from Virtual Work

From Newton's second law, $f=ma=m{\ddot x}$ (introduced in Eq.(B.1)), we can use d'Alembert's idea of virtual work to 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,

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

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 $d(m\,v^2/2)$ . The kinetic energy of a mass moving at speed $v$ is then given by the integral of all such differential boosts from 0 to $v$ :

$$E_m(v) = \int_0^v dW = \int_0^v d\left(\frac{1}{2}m \nu^2\right) = \frac{1}{2}m v^2 = \frac{1}{2}m\,{\dot x}^2,$$

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

The quantity $dW=f\,dx$ is classically called the virtual work associated with force $f$ , and $d x$ a virtual displacement [548].