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Given that one has a (differentiable) function $a(x)$ which gives a particle's acceleration at a certain position, $x$, how would one go about finding the particle's velocity at that position assuming the particle started with zero velocity at $x=0$?

More importantly, how would one link the equation to time? I know it sounds silly, trying to make a variable appear out of nowhere, but I can't see how, given the acceleration at any point, one cannot tell where the particle is at a certain time.

Edit: I thought I should add, this is just a thought that occurred to me not an actual (textbook or otherwise practical) question so I'm afraid there isn't much more information I can give.

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Use the form of acceleration $$a=v\frac{dv}{dx}$$ and solve the differential equation to get $v$ in terms of $x$. Then use $v=\frac{dx}{dt}$....

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  • $\begingroup$ I loved it when I first saw my professor do this in class. $\endgroup$
    – Plopperzz
    Feb 14, 2017 at 22:07
  • $\begingroup$ You are welcome $\endgroup$ Feb 15, 2017 at 6:26

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