(covers information from several alternate timelines)
Sometime during the early to mid-2240s of the alternate reality, James T. Kirk's stepfather owned a vintage red 1965 Corvette Sting Ray C2 convertible. Kirk went for a joyride in the car, was chased by a police officer on a hovercruiser, and bailed out just as the car fell into a massive quarry. (Star Trek)
Background information Edit
Triple vertical brake vents (behind the front wheels) identify the car in the film Star Trek as a 1965/1966 Corvette C2. The lack of the "Corvette" name on the left front hood further identifies it as a 1965 model.
In a deleted scene from Star Trek, it is revealed that the 1965 Corvette Stingray actually belonged to George Kirk. In the aftermath of George's death, Jim and George Jr.'s uncle, Frank, treated the car as his own. Jim found the keys to the car while washing it, and drove off with it in an act of defiance against the abusive Frank.
A critic for Slashfilm.com suggested that the destruction of a 1960s model Corvette symbolized J.J. Abrams jettisoning the original incarnation of Star Trek, commenting, "Is it just a coincidence that the car was released during the same decade that the original Star Trek television series premiered? I believe that by throwing the 1960's era car off a cliff, Abrams was making a statement – 'this is not your father's Star Trek movie – we're throwing all that stuff away, off a cliff no less. This is the new Star Trek."  However, Abrams himself denied that this was his intention, remarking, "The idea was to show the renegade, young Kirk and have a wildly anachronistic scene where you had an earthbound, almost back-looking scene combined with a forward-looking futuristic scene technologically. It had nothing to do with that kind of metaphor." 
In the film's novelization, the Corvette had been a prized possession of Kirk's father, and young Jim – who had been washing it – embarked on the joyride after being told by his own brother that their stepfather planned to sell the car, behind their mother's back, while she was off-planet.