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If we consider a damper (dashpot) element that exerts a force opposite the direction of motion proportional to the velocity, i.e. $$ \vec{F} = -c \vec{v}$$

Therefore, we can consider an infinitesimal amount of work done on the damper fluid, $$ dW = F \, dx = (c \dot{x}) \, dx $$

And the time rate of work done on the damper fluid to be, by dividing by $dt$, $$ \frac{dW}{dt} = c \dot{x} \frac{dx}{dt} = c \dot{x}^2 $$

While this makes sense (and can be confirmed within any Dynamics Textbook), I'm uncomfortable with just "dividing by $dt$". How is this legitimate? Is there an alternative way to derive this same thing?

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    $\begingroup$ The "dividing by dt" step is entirely legitimate, because it is not dividing by dt. Let me refer you to this related answer I gave someone a while back which might be helpful in putting you at ease here $\endgroup$
    – Lourenco Entrudo
    Commented Nov 21, 2023 at 22:39
  • $\begingroup$ Write it like this $~ dW=c\,\dfrac{dx}{dt}\,\dfrac{dx}{dt}\,dt$ $\endgroup$
    – Eli
    Commented Nov 22, 2023 at 16:37
  • $\begingroup$ @Eli, how so? Wouldn't it be: $$dW = c \frac{dx}{dt} dt$$ Either way, how would that change anything? $\endgroup$
    – Jacob Ivanov
    Commented Nov 23, 2023 at 2:19

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