GA pilot reported near misses with two aircraft during final approach and landing to VPC non-towered airport requiring a go-around from one of the other aircraft to avoid a possible collision. Reporter stated the faster approach speeds of the other aircraft contributed to the event.

Narrative

In the traffic pattern at VPC (Cartersville) teaching takeoff and landings Aircraft Y and Aircraft Z both reported on a long straight in to Runway 19. My aircraft reported downwind and some time after that the Aircraft Y reported a 6 mile final. In order to avoid Aircraft Y and Aircraft Z behind him I instructed my student to make a short approach (Left turn to base) and reported left base. I believe it was about that time that I heard the Aircraft Y report a 4 mile final. In had hoped to order to avoid extending the downwind into a potential head-on conflict with Aircraft Z that had also reported a long final behind the Aircraft Y and I had instructed my student to make the short approach. The closure rate of the Aircraft Y was much faster than I had anticipated and the two aircraft came into conflict after the time my student turned final. By making a short approach my intent was to avoid a conflict with both aircraft on long final straight in and avoid evasive maneuvers in the pattern.I was hesitant to make right 360's in lieu of turning base to final since there were two aircraft on straight ins and I was concerned that repeated 360's could create a conflict with any other aircraft that might be on downwind behind us (I assumed we would need at least 2 360's). I didn't want a head-on conflict with Aircraft Z by extending downwind not knowing if Aircraft Z was right or left of centerline on his straight in and I was hesitant to turn right to fly out of the pattern into what would be opposite direction to any traffic on a wider base (none had announced that position that I was aware of but it seemed like a bad idea).When we announced final I heard what I thought was the Aircraft Y asking what he should do. I repeated that we were short final and that was up to him. I wasn't trying to be snarky; but since our back was to him I wasn't in a position to direct him.Aircraft Y went around and over top of us.Trying to avoid the two aircraft on long straight put us in a difficult position to avoid both these aircraft and continue to avoid creating a conflict with any other aircraft that might be in the pattern that I might not be aware of.In retrospect I could have possibly taken control of the aircraft and turned right (east - opposite direction to the base leg) instead of turning left to base; started a climb back to and above pattern altitude and flown a few miles east of the airport to re-enter downwind on a 45.I did the best I could given the information I had and my assumption that Aircraft Y's final approach speed would allow us to avoid any conflict or unusual evasive action by making a short approach. It seemed like the best option at the time. Aircraft Y's final approach speed and closure rate were simply much faster than I anticipated and there was a conflict on our short final just before touchdown as he flew over top of us.