Including version number in the URL is the standard approach as I explained in this post (I do not repeat the content): Implementing versioning a RESTful API with WCF or ASP.Net Web Api
You do not need to create a completely new project although you can. The problem that you will be facing with a single project is that there will be collision of names:
/api/v1.0/Car/123
and
/api/v2.0/Car/123
both will point to CarController
while you can have only one of those. The solution would be to implement your own IHttpControllerSelector
and register with the DependencyResolver. This implementation will look at the version number and perhaps find the type based on the namespace.
UPDATE
I do not intend to start a REST controversy here. But as @DarrelMiller points out, here is an older discussion on the same subject discouraging my suggested approach:
How to version REST URIs
I personally think URL versioning is the way to go.