The flight instructor reported that the student pilot was landing on the 1,760-ft-long runway and flared above the numbers. The airplane floated halfway down the runway and he instructed the student pilot to initiate a go around. With about 1/4 of the runway remaining, the flight instructor was concerned the airplane would not have sufficient airspeed to climb above the tree line. He took control of the airplane and landed in the grass field past the end of the runway. The airplane came to a stop in the mud, the nose gear separated, and the airplane sustained substantial damage to both wings and the firewall. The student pilot reported that, the airplane was "climbing out normally," when "without any warning" the flight instructor took the flight controls. He further reported that, the instructor "completely lost control and was very excited." Airport surveillance video showed the airplane begin to climb out, then abruptly descend towards the runway. The flight instructor further reported that there were no preaccident mechanical failures or malfunctions with the airplane that would have precluded normal operation.