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In a derivation for absolute energy equation for a structure subjected to an earthquake, the following equation appears:

$\int m \frac{d\dot v}{dt}dv =\frac{m(\dot v)^2}{2}$

Would someone please elaborate on the intermediate steps? I dont understand how we can go from the left hand side to the right hand side of this equation. We have $m$ is constant and $v$ is a function of $t$.

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  • $\begingroup$ What is the derivative $\frac{d(\dot v)^2}{dt}$? Or can you integrate by parts? $\endgroup$ – Mark Viola Nov 14 '15 at 22:56
  • $\begingroup$ It is $\frac{d}{dt} (\frac{dv}{dt})^2$ $\endgroup$ – user32882 Nov 14 '15 at 22:58
  • $\begingroup$ It's $2\dot v \frac{d\dot v}{dt}$. $\endgroup$ – Mark Viola Nov 14 '15 at 23:00
  • $\begingroup$ but the original integral is with respect to $v$... $\endgroup$ – user32882 Nov 14 '15 at 23:05
  • $\begingroup$ $dv \to \dot v\,dt$ $\endgroup$ – Mark Viola Nov 14 '15 at 23:09
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Let $v=v(t)$. Then $dv=\dot v\,dt$ and we can write

$$\begin{align} \int m\frac{d\dot v}{dt}dv&=m\int \frac{d\dot v}{dt}\dot v\,dt\\\\ &=\frac12 m\int \frac{d(\dot v)^2}{dt}\,dt\\\\ &=\frac{m(\dot v)^2}{2} \end{align}$$

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