When actually making contact with the car, make sure you're pretty much just pushing it's back end out, not smashing into it because you can't do any damage to the vehicle for gold. Once it's gone all of the way around you'll need to pull to the right and try to stop with the front of your car touching the back of the other one. If you do it right, the car will spin around right in front of you, and you'll need to use your car to keep the car spinning around by holding left as you accelerate. Speed to the left side of the speeding police car, then just as the front of your car gets level with the back tire of the speeding car, make a sharp right and hit the tail end of the vehicle.
Watch the short movie before starting the mission and you'll see just what you're going to do. Finish as close to the other car as possible.
Took me various attempts to achieve it and thus complete this great game later. Maneuver to spin the other car around with minimum damage. Answer: If I can remember it was a mission later in game and was a tricky one indeed.
