All planes are equipped with a device called a transponder.   The transponder emits a signal, picked up by the radar, that allows the plane to be identified as a plane.  When a pilot contacts an FAA radar facility, the radar facility people issue the pilot a trasponder code, which the pilot types into his transponder so that it emits that code, identifying the plane on the radar screen as the individual who contacted the tower.  They know exactly who the blip on the screen is.  Kennedy was flying VFR (Visual Flight Rules) and was not required to contact the tower.  In addition, when you are given the transponder code, that also activates a computer system that triggers a "low altitude alert" if your plane flies below 100 feet anywhere but over an airport runway.  So that the FAA would have received a computer generated low altitude alert before Kennedy's plane hit the water.
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