Finding the velocity of an object, given its acceleration as a function of position

Question

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.

Answer 1

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