The flight instructor reported that, while the airplane was in the landing flare for a touch-and-go landing, the student pilot allowed the airspeed to become too low, and he instructed the student to perform a go around. The student applied full power and retracted the flaps, but the airplane continued to descend, so the instructor "got on the controls with him." During the go-around, the airplane touched down on the runway hard, bounced, and the instructor reported hearing a "ping" and observed that the right main landing gear was no longer attached. The instructor continued the go-around and entered the traffic pattern; during the approach, he observed the right main landing gear laying on the right side of the runway. He landed the airplane and it veered to the right off of the runway. The airplane sustained substantial damage to the fuselage. The flight instructor reported that there were no preaccident mechanical failures or malfunctions with the airplane that would have precluded normal operation. The automated weather observation station located on the airport reported that, about 36 minutes before the accident, the wind was from 250° at 9 knots. The airplane landed on runway 34.