I think he is talking about something miniature (pulling car by string) ? That would greatly simplify the cost/scope/time.
Not that I have done anything remotely like this, but ... I have seen numerous different approaches to following...
I understand that some camera systems can help with tracking of a color (CMUcam). Or perhaps use some kind of beacon (IR pair) ? Perhaps you can use a different system to detect and maintain a constant distance?
I think 4 months is doable depending on scope and how much you know, how much time you have to work on it, etc.
Projects like this are easier, at least in my experience, when you can document
what you want it to do (requirements), then worry about
how to get it to do those things later (architecture & design).
Also easier if you can sort out the requirements into logical functions -- basically breaking down the problem into smaller problems that you can independently solve more easily. E.g., the "car" has to accelerate, decelerate, and steer. The functional chunks are "speed control" and "steering". Another function is "track and steer" -- detecting the other car and what direction it is in so you can steer correctly. Another function is "maintain distance" -- measuring distance and altering speed accordingly.
Hope this helps
Michael