Well here’s what we did, though this response is a couple years too late for your project it may assist someone else.
After some test shooting we discovered.
1. You don’t need to go “fast”
2. It can be accomplished affordably and easily and most of all legally (depending on interpretation of the cop) If a cop stops you record it, then ask if he’ll be in the film for you. If you aren’t breaking any laws then he’s stopping you out of curiosity. If he’s game set up all your cameras on him going down the street from various angles.
Here’s what we did. We utilized 2 Hero GoPro cameras, a Sony FX-1, and a Sony HDR-CX150. We opted for multiple cameras to reduce the number of times we needed to go through the sequence and still required 6-8 set ups to acquire the shots we needed.
After determining the number of cars needed, collecting the cars and drivers we had about a two hour meeting, going over the route, what the cars would do, when and how this would be executed. Coordination is very important.
Once on location we got set up. the FX-1 mounted in a “chase car”, the GoPro’s set up on lower fenders, quarter panels, hoods, trunk lids at various angles and the CX-150 for “in car” shots. We’d set up the cameras on the cars as needed gaining 3-4 POV’s from each car for the sequence, set up the chase car in front or behind to get master shots. After running through this several times we also set up the FX-1 & CX-150 road side, eye level & curb level for another run through.
Mind you we never exceeded the speed limits, 35mph-45mph on a fender shot has a huge sensation of speed and close distance to other traffic, as does utilizing wide angles from hoods & trunks. More objects pass the lens in a short time frame. Tight Zoomed shots on specific vehicles also accomplishes this result, especially when panning.
Now comes the hard work, editing. The edit requires lots of intercuts, Foley and jump cuts. Be sure to get shots of Tachometers revving, Speedometers at high speed (a decent animator can fake this), clutches getting pushed/released, accelerator pedals pushed, brake pedals pushed, shifting gears etc. If any of the chase shots appear too slow they can be accelerated/zoomed/reverse zoomed in your software. The success of your car chase will be in presenting a sensation of speed, action & close proximity which, once you have your footage, can be accomplished in the edit.
Good Luck & Be Safe