How Fast Do Point Charges Move After Being Released?

[ELB27]

Homework Statement

Two positive point charges, $q_A$ and $q_B$ (masses $m_A$ and $m_B$) are at rest, held together by a massless string of length $a$. Now the string is cut, and the particles fly off in opposite directions. How fast is each one going, when they are far apart? [from 4th edition of Introduction to electrodynamics by Griffiths]

Homework Equations

General definition of work done from point $a$ to point $b$: $$W = \int_a^b \vec{F}\cdot \vec{dl}$$
Electric potential energy stored in a system: $$W = QV$$ where $V$ is the potential of the system.
Electric potential due to a positive point charge $q$: $$V = \frac{1}{4\pi\epsilon_0}\frac{q}{r}$$ where $r$ is the distance to the charge.
Gravitational potential energy due to a point mass $m$: $$U = -\frac{Gm_1m_2}{r}$$ where $G$ is the gravitational constant and the sign difference due to the electric force of a positive charge being repulsive while the gravitational force always attractive.

The Attempt at a Solution

My idea is to first add the two potential energies of the system (electric and gravitational) using as reference point $\infty$ which will give the total potential energy stored in the system while it's still at rest. Then when they are "far apart" (as I understand it: at infinity), all of the aforementioned potential energy will be converted to the kinetic energy of the two charges/masses, giving me the first equation. For the second equation I used the conservation of momentum of the system consisting of the two charges/masses and the fact that the initial momentum was zero. Thus I get two equations in two unknowns:
$$\begin{cases} (1) \frac{1}{a}\left(\frac{1}{4\pi\epsilon_0}q_Aq_B - Gm_Am_B\right) = \frac{1}{2}\left(m_Av_A^2 + m_Bv_B^2\right)\\(2) m_Av_A = m_Bv_B \end{cases}$$
Solving (1) and (2) for $v_A$ and $v_B$ I get:
$$v_A=\sqrt{\frac{m_B}{m_A(m_A+m_B)}\frac{2}{a}\left(\frac{1}{4\pi \epsilon_0}q_Aq_B - Gm_Am_B\right)}$$
$$v_B=\sqrt{\frac{m_A}{m_B(m_A+m_B)}\frac{2}{a}\left(\frac{1}{4\pi \epsilon_0}q_Aq_B - Gm_Am_B\right)}$$

Are my reasoning and answer correct? In particular, have I accounted for the fact that both charges move simultaneously and with different speeds, resulting in awkwardly changing electric and gravitational fields as they fly apart?

Any feedback will be highly appreciated!

EDIT: I am also interested in a systematic way of checking my answers by my self. I am too used to using answers sheet and I want to get rid of this habit of mine. Any suggestions on this matter?

 

[haruspex]

Your method and answer look right.
I sanity-check dimensional consistency and boundary cases. E.g., mA = mB, mA almost zero, ...

 

[ELB27]

Your method and answer look right.
I sanity-check dimensional consistency and boundary cases. E.g., mA = mB, mA almost zero, ...

Thanks for the answer! Any other suggestions?

 

[haruspex]

Thanks for the answer! Any other suggestions?

Symmetry preservation is another test.

 

[ELB27]

Symmetry preservation is another test.

OK, thank you very much, I'll keep these methods in mind!