To solve for the velocities, we need another relationship between them. Because no external forces act on the system, momentum will also be conserved. Before the string is cut, the momentum is zero, so the momentum has to be zero all the way along. Potential plays the same role for charge that pressure does for fluids. If there is a pressure difference between two ends of a pipe filled with fluid, the fluid will flow from the high pressure end towards the lower pressure end.
Charges respond to differences in potential in a similar way.
Electric potential is a measure of the potential energy per unit charge. If you know the potential at a point, and you then place a charge at that point, the potential energy associated with that charge in that potential is simply the charge multiplied by the potential. Electric potential, like potential energy, is a scalar, not a vector. These often appear on field line diagrams. Equipotential lines are always perpendicular to field lines, and therefore perpendicular to the force experienced by a charge in the field.