When two charges move under their mutual attraction or repulsion, the Coulomb force is an internal force to the system. And there is a principle, valid for any system of particles, that internal forces cannot violate:
Principle — The motion of the centre of mass
The centre of mass of a system moves as if all internal forces did not exist, under the action of external forces alone. The Coulomb force between the two charges, however intense, does not shift the trajectory of the centre of mass by a millimetre.
A neat and very convenient separation follows. If the system is acted on, say, only by Earth’s gravity (an external force), the centre of mass traces an ideal parabola, exactly like a point-like projectile: its maximum height, its time of flight, its velocity are calculated by completely ignoring the electric interaction between the two charges. It is only when you want to know the velocity of the individual charges that electrostatic potential energy comes into play.
This suggests the centre-of-mass trick for collision problems with conservative forces (Coulomb, springs): the total kinetic energy is split into centre-of-mass energy plus internal energy (König’s theorem),
and one notes that the motion of the CM is trivial, while all the “physics” of the interaction lies in the internal energy. A question like “how close do the two charges get to each other?” then translates into the condition : the point of closest approach is where, in the centre-of-mass frame, the two particles are momentarily at rest and all the internal energy has become electrostatic potential energy. A two-body problem is thus reduced to a simple one-body energy balance.
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Topics: Electrostatics Concepts: Electrostatic potential energy · Conservation of momentum Methods: König’s theorem
Related exercises: Problem — Two charges released from rest · Problem — The two gunslingers · Problem — Two charges launched upward