While on the final leg of approach for landing, the pilot of the Cessna 182T stated that the airplane was "about four or five knots faster than usual." She reported that there were no wind gusts on the approach, there was no turbulence, and that the airplane felt "smooth." During the landing, the airplane touched down on the runway once and bounced. The second time the airplane made contact; the pilot heard a crack and knew something was wrong. The pilot increased engine power slightly to avoid stalling, but the airplane subsequently stalled. The pilot noted the stall did not feel the same as a practiced stall, and the airplane came down "hard" on the runway. The pilot believed the airplane landed on the main landing gear the first touchdown, and landed on the nose wheel the second touchdown. The nose gear collapsed, resulting in substantial damage to the airplane.