Two small equally charged spheres, each of mass $m$, are suspended from the same point by silk threads of length $\ell$. The distance between the spheres is $x\ll \ell$. Find the rate $dq/dt$ with which the charge leaks off each sphere if their approach velocity varies as $v=a/\sqrt{x}$, where $a$ is a constant.
This is question 3.3 from IE Irodov's electrostatics part. The solution of this question comes out to be $$x^3=\frac{\ell q^2}{2\pi mg\epsilon_0}, \text{ } \frac{dq}{dt}=\frac32a\sqrt{\frac{2\pi mg\epsilon_0}{\ell}}$$
Now this solution comes under the assumptions that the charges are always under equilibrium.
I have following confusion:-
If the relative velocity of approach is $v=a/\sqrt{x}$ is a function of $x$, that implies there is acceleration which means that our assumption of charges being in the equilibrium is wrong.
What is the explanation for the assumption of equilibrium condition?
