Solving dynamics of two oppositely charged particles without using energy conservation Two charges, each $+q$, are separated by distance $d$.  One of the charges is fixed, and the other is free to move under the influence of the force from the first.  The second particle is initially at rest.  
Can this be solved without using arguments about conservation of energy?  I understand how to find the acceleration as a function of distance.  I don't know how to express the acceleration as a function of time to integrate to find velocity and position. 
 A: A way to get a function with respect to time:
You already have an equation of acceleration and displacement as you've mentioned(you got it from applying Coulomb's Law), hence you have an equation in the form:
$$a=\frac{k}{x^2}$$
Where $a$ is acceleration and $x$ is displacement and $k$ are the constants involved. Now notice that $a$ is the second-derivative of $x$ with respect to time, so $a=\frac{d^2x}{dt^2}$. Hence the equation becomes:
$$\frac{d^2x}{dt^2}=\frac{k}{x^2}$$
This is a quite complicated second-order differential equation and you can solve it(detailed steps in the second answer here) using the initial conditions to get displacement in terms of time x in terms of t and hence find velocity in terms of time.  Notice that this method is much more cumbersome than using energy conservation especially if we know just want to know suppose the speed at a certain distance. Thus, of-course this problem can be solved with-out energy conservation but it is much easier to usually solve it with using energy-conservation. 
