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If this takes place in location to which you have easy access, I recommend shooting practical effects. For dialogue, put the car on a low trailer. Mount a couple-three GoPros for the angles you want.
For non-dialogue shots, jount the GoPros on various spots on the car and drive the same route.
The result will be a lot better than faking it.
To make the wheels move you need to add some fake motion blur to the wheels. Create a mask on an adjustment layer and add radial blur centered on each wheel. If the shot is wide enough you can probably pull it off with just some simulated motion blur. Do the same with the road but use directional blur and maybe add moving road shot masked and composited with multiply to give the pavement some motion.
If the shot is close enough and you want to see the rims turning you'll need to shoot completely flat lit shots, stills will work, of the wheels that you can mask and turn into spinning wheels. You then overlay the spinning wheels on the actual wheels, distort them as needed to get the flat part that touches the ground, and then experiment with blend modes like Multiply so the lighting effects on the real tires are added to the spinning wheels.
If there are background elements that need to pass behind the car you'll need to do some roto. Shooting green screen to simulate night and a car will require either a big location or a big green screen far enough behind the car to give you a chance to properly light the car so it looks like a night shot.
If the shot in your example 2:41 - 2:43 is the shot you are talking about I'd just use the radial blur trick and add a masked moving road shot multiplied over the static road and be done with it. Some pretty subtle enhancements to the shot would help sell it if you really feel the need.