The good news is that cars do not travel at the speed of light.
Only massless particles, like photons, may travel at the speed of
light, and in fact massless particles may travel at no other speed. An
object which has some mass, like a car, requires ever more energy to
accelerate it to speeds approaching the speed of light, and no matter
how much energy you give the car, it will always travel at a speed less
than the speed of light.
The question might then be what would the situation be like if the
car were going at 0.9999999 times the speed of light? If you are in the
car, and look only at stuff which is traveling along with you at your
speed, you would not notice the difference, and the headlights will
work normally. If another car is ahead of you traveling at the same
speed, then your headlights will light up that other car and you will
see it normally. This happens because the laws of physics are the same
in all inertial frames of reference (where "intertial" means "moving
uniformly without accelerating". Rotating also involves acceleration,
so the frame of reference cannot turn).
The road might look strange, however. It will appear to be
shortened (Lorentz contracton), and the photons from your headlights
will bounce off of the road and back at you with very large energies
because of blueshifting. You may not see your taillights reflected on
the road either (they will be redshifted), but the car behind you
traveling at the same speed will be illuminated by your taillights in
the normal way.
If you are not in the car but are standing at the side of the
road, then the headlights of the car will make a beam of very
high-energy photons (maybe x-rays, maybe gamma rays, depending on how
fast the car is going). The light rays from the headlights will
actually not spread out but will become more directed along the line of
motion of the car because the component of the momentum of the photons
along the car's direction will increase due to blueshifting but the
transverse components will stay the same.
(published on 10/22/2007)