Hello everyone, this is my first post to PPRUNE. I'm a private pilot,
however during the day I'm a particle physicist, so feel qualified to
respond.
Firstly, as has alread been stated, it will not be possible to move at EXACTLY the speed of light. The inertial mass of an object (that is the mass that you think of when you calculate momentum, kinetic energy, etc) increases with velocity and as a result an infinite amount of energy is required to accelerate an object to exactly the speed of light. Please note that this is the mass of the "car" as viewed by a stationary external observer. From the point of view of the driver, the car is stationary and so the mass has not changed at all. Hence, arguments about being unable to open your eyelids are not valid. I will return to time dilation, length contraction, etc later.
As has already been stated, the argument can still be made for a hypothetical car moving at 90%, 99%, etc of the speed of light.
One of the important points of Special Relativity is that the speed of light is constant irrespective of the frame of reference, so from the point of view of the car, the light is moving at the speed of light. From the point of view of an observer at the side of the road the light is STILL moving at the speed of light.
So, for our car at 90% of the speed of light, the light will indeed strike the road and some fraction of it will be reflected back to the driver.
The driver will then see this light, which has been reflected by the road which to the driver is racing towards him at 90% of the speed of light. As a result, the light will be "blue-shifted" and depending on the speed of the car, it will arrive with a much higher frequency and thus not be visible to the driver's eyes.
To return to the comments about time dilation and length contraction. It should be remembered that these are what an external observer sees when an object travels close to the speed of light.
So, the driver in the car, looks at his/her watch and sees the second hand moving at the same speed as before. To the external observer, however, the clock will appear to be running very slowly. This is the origin of the twin paradox (a google search should find this easily enough for those who have not encountered it).
The length of an object appears to be contracted IN THE DIRECTION of that object's motion.
So from the point of view of an external observer, the car will appear to get shorter. The driver (sitting in their seat) will be the same height, but their arms reaching to the steering wheel will appear (to this external observer) to be shorter.
From the point of view of the driver, they have not changed one bit, however the road, which is rushing towards them at 90% of the speed of light, will appear to be shorter.
These results of Special Relativity are confirmed (and indeed
used) by physicists such as myself in our day-to-day work.
To bring this into a slightly aviation-related context, a large fraction of the cosmic rays that are produced in the upper atmosphere and arrive at the Earth's surface are elementary particles called muons.
A muon is like a heavy electron, except that its lifetime is 2 micro-seconds!
A quick calculation shows that the muon should thus travel (on average) 600 metres. This cannot possibly be the case as the muons are produced in the very upper atmosphere.
So, from the point of view of an observer on earth, you can say that the muon, travelling close to the speed of light, travels further becaus its internal clock is slowed down, and thus it does decay after 2 microseconds, but that 2 microseconds takes a lot longer to occur for the observer on Earth.
From the point of view of the muon, the Earth is rushing towards it at close to the speed of light, and so the distance it has to travel is contracted. The muon thinks that is has lived for 2 microseconds, but that the size of the entire atmosphere has shrunk to less than 600 metres!
I hope this helps.