I don’t know of a perfect solution, and I have a feeling that one does not exist. Even if you could accurately predict the future position of the vehicle, you would be unable to predict the way the user will operate the controls. So the problem comes down to minimizing the negative effects of client/server lag. With that in mind, I would approach this from the position of the principle of least astonishment (paraphrased from Wikipedia):
In user interface design, the principle of least astonishment (or surprise) states that, when two elements of an interface conflict, or are ambiguous, the behaviour should be that which will least surprise the human user at the time the conflict arises.
In your example, each user sees two vehicles. Their own, and that of another player. The user expects their own vehicle to behave exactly the way they control it, so we are unable to play with that aspect of the simulation. However, the user can not know exactly how the other user is controlling their vehicle, and I would use this ambiguity to hide the lag from the user.
Here is the basic idea:
- The server has to make the decision about an impending collision. The collision detection algorithm doesn’t have to be 100% perfect, it just has to be close enough to avoid obvious inconsistencies.
- Once the server has determined that two vehicles will collide, it sends each of the two users a message indicating that a collision is imminent.
- On client A, the position of vehicle B is adjusted (realistically) to guarantee that the collision occurs.
- On client B, the position of vehicle A is adjusted (realistically) to guarantee that the collision occurs.
- During the aftermath of the collision, the position of each vehicle can be adjusted, as necessary, so that the end result is in keeping with the rest of the game. This part is exactly what MedicineMan proposed in his answer.
In this way, each user is still in complete control of their own vehicle. When the collision occurs, it will not be unexpected. Each user will see the other vehicle move towards them, and they will still have the feeling of a real-time simulation. The nice thing is that this method reacts well in low-lag conditions. If both clients have low-latency connections to the server, the amount of adjustment will be small. The end result will, of course, get worse as the lag increases, but that is unavoidable. If someone is playing a fast-paced action game over a connection with several seconds worth of lag, they simply aren’t going to get the full exeperience.