Change
private ArrayList finishingOrder;
//Make an ArrayList to hold RaceCar objects to determine winners
finishingOrder = Collections.synchronizedCollection(new ArrayList(numberOfRaceCars)
to
private List finishingOrder;
//Make an ArrayList to hold RaceCar objects to determine winners
finishingOrder = Collections.synchronizedList(new ArrayList(numberOfRaceCars)
List is a supertype of ArrayList so you need to specify that.
Otherwise, what you're doing seems fine. Other option is you can use Vector, which is synchronized, but this is probably what I would do.