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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,

f(t) &=& m\, {\ddot x}(t)\\
\,\,\Rightarrow\,\,\quad d W\isde...
...{1}{2}{\dot x}^2\right)\\
&=& d\left(\frac{1}{2}m\,v^2\right).
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$:

$\displaystyle 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 [544].
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