Electric interaction between two negatively charged particles of different sizes on a mixed background of positive, negative, and neutral particles is complex and has relevance both to dusty plasmas and to transports in ionized fluids in general. We consider particularly effects during interaction that particle velocity and neutrals in the background may have on the well-known “dressing” and electric shielding that is due to the charged part of the background and how the interaction energy is modified because of this. Without such effects earlier works show the interaction becomes attractive when the distance between the two particles is a bit larger than the Debye length. We use a model where one of the two interacting particles has a radius much larger than the Debye length and the other a radius shorter than the Debye length. Then, the complex interaction may be more easily determined for particle separation up to a few Debye lengths. We consider the larger particle as stationary while the smaller may move. We find quite simple analytic expressions for the dressed particle interaction energy over the whole range of speed of the incoming smaller particle, assumed coming head on the larger particle, and the whole range of neutral particle densities. We also derive a distance of closest approach of small and large particles for all such parameter values. This distance is important for excluded volume estimations for moving small charged particles in media populated by large charged particles on a background as described above, and hence, important for determining the speed of flow of the smaller particles through such media.