Kinetic energy of two charged balls at infinite distance between them If I have two balls with masses and charges $m_1, q_1^{+}$, $m_2, q_2^{+}$, initially held at distance $d$, and then released, how can I know the kinetic energies of each of the balls at infinite distance between them? I'm quite stuck on that, because they both have the same potential energy at the beginning, and it decreases not in the same pattern, as if one of the balls was stationary. So it not only falls like $1/R$, because at the same time, the other ball that is causing this potential energy, is also being repelled. So how can I really find out the energies? I tried to apply the conservation of energy law, because I know that at infinite distance from each other they'll have zero potential energy, thus all the initial was transformed into kinetic form, however I'm stuck with the initial potential energy (they both have it, so should I put $2U_p$?), and even so, I can't find their kinetic energies separately, without having another equation.
 A: Potential energy is a property of the system, not any one object. Thus there should only be one copy of the typical $1/r$ potential energy between two charges (plus an analogous gravitational term if that can't be neglected).
The easiest way to see this is to start from "infinite" separation. Instead of pushing the two charges together, hold one fixed and move the other toward it. The moving charge must fight the standard Coulomb force (with a little help from gravity) to get closer to the stationary one, so the potential energy obtained here is just the integral of this force over the distance traversed ($d$ to $\infty$).
But what about the stationary object? Well, sure, we need to exert a force on it to keep it from being repelled by the approaching charge. But it is not moving, so the change in $\vec{F} \cdot \vec{x}$ energy vanishes.
The fact that at some point in the future we will let both objects move doesn't change the potential energy, so you should get the same potential energy as if the problem were stated:

A point mass $m_1$ with charge $q_1$ is fixed at the origin. Another point mass $m_2$ with charge $q_2$ is brought in from infinity. What is the potential energy of the system?

It may also help to remember that "$2\infty = \infty$." Moving objects from $x = -\infty$ and $x = \infty$ to the origin covers the same distance as moving one object from $x = \infty$ to the origin.
A: I would suggest you to use this equation:
$$
W=\int_C{Fdx},
$$
where $F$ is the force on an object and $W$ the work done by this force.
In this case there are two types of forces acting on the two objects, gravitation and Coulomb force:
$$
F_{result}=\frac{1}{4\pi\epsilon_0}\frac{q_1q_2}{r^2}-G\frac{m_1m_2}{r^2}=\left(\frac{q_1q_2}{4\pi\epsilon_0}-Gm_1m_2\right)\frac{1}{r^2},
$$
with $r$ the distance between the two objects. So:$$
W_{total}=\left(\frac{q_1q_2}{4\pi\epsilon_0}-Gm_1m_2\right)\int^\infty_{d}{\frac{1}{r^2}dr}=\left(\frac{q_1q_2}{4\pi\epsilon_0}-Gm_1m_2\right)\frac{1}{d}
$$
Edit: This isn't entirely correct, since I am assuming that this is a symmetric situation, so $m_1=m_2$ and therefore $W_1=W_2=\frac{W_{total}}{2}$. This will affect the ratio of distance to the origin of the two objects and therefore the amount of work done on each object.
