On August 6, 1993, at 1515 central daylight time (CDT), a Piper PA-28R-200, N233LC, registered to the Skyhawk Flying Club, Incorporated, of Omaha, Nebraska, and piloted by a private pilot, was substantially damaged during a forced landing shortly after takeoff. Visual meteorological conditions prevailed at the time of the accident. The personal 14 CFR Part 91 flight was not operating on a flight plan. The pilot and three passengers reported no injuries. The flight originated from Millard, Nebraska, at 1515 CDT. The pilot stated the airplane's engine sound diminished along with its climb performance shortly after takeoff. The airplane began a descent shortly after it began the initial climb and struck a soccer field goal post prior to colliding with the ground. One witness stated she did not hear N233LC's engine as it was descending. This witness observed the airplane collide with the soccer field goal post and ground. A second witness observed N233LC "...flying very low..." and collide with the goal post and ground. The on-scene investigation was conducted by a Federal Aviation Administration Principal Maintenance Inspector (PMI). N233LC's engine was removed from the airframe and test run. The test run revealed the engine was capable of operating to the manufacturer's specifications. The PMI examined the fuel distribution and exhaust systems for damage which could result in an engine power loss. He stated he did not find anomalies within the systems. The PMI observed an unattached alternate air door (door) inside the air induction system. The door was found lodged in the alternate air adapter assembly. According to the PMI, "The Piper PA-28R-180 and early model PA- 28R-200 Arrows incorporated an alternate air door on the cover of the induction air box assembly. The door, P/N 23809-07, which was in the induction system of this aircraft was for the earlier model PA-28R-200 Arrows and should not have been installed on this aircraft. The air box cover on this aircraft was replaced in 1986 with a P/N 67745-00 which is the correct part number for this aircraft." The PMI stated the previously installed incorrect door had broken off and was ingested downstream in the induction system. The PMI's examination statement is appended to this report. Maintenance records for N233LC reveal the installation of a "...new induction air filter box cover," on November 10, 1986. The airplane had been inspected over a period of seven years without the incorrect door being found in the induction system. Pertinent excerpts of the maintenance records are appended to this report. N233LC's propeller hub, governor, and blades were examined. The result of the examination found the propeller hub and governor functioned to manufacturer's specifications. The propeller tear down reports are appended to this report.