Length and Distance
Units of Measure

If a car is traveling at the speed of light and its headlights turn on would they work?



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yes the photons light is made of would be moving at the speed of the car plus the speed light travels on its own. this follows the same rule as to why anything dropped out of a moving object follows the path of the object it is dropped from. The above "yes the photons light is. . ." answer is wrong.
First of all, you need to realize that when you say you are traveling at the speed of light, that has to be with respect to, or relative to, something else. It is an underlying fundamental assumption of Einstein's special theory of relativity that uniform, non-accelerated motion has no meaning of and by itself. That is, there is, by assumption, no meaning to the idea of moving uniformly at the speed of light in an empty universe. That state is completely equivalent to being at rest in an empty universe..

I preface my answer with this comment because it leads immediately to the answer to the question. Imagine that you are in your car 'traveling at the speed of light' and that you turn on your headlights. That state of motion is utterly equivalent to being at rest in an empty universe. Since, when at rest, the light from your headlights would be launched forward from your car at the speed of light, relative to you, with a certain color spectrum, that is exactly what would happen if somehow you could be moving instead at the speed of light.

In other words, the presence or absence of other objects or matter in the universe relative to which, if present, you could make a determination that you were moving at the speed of light makes absolutely no difference to your own experiences and experiments. The light that you launch behaves in exactly the same way whether the other referential matter exists or not.

This leads into another interesting question, however. And that is whether the rest of the matter (mass) in the universe in some way affects your own local observations. So far this question has come up in relation to theories of gravity. If effect, the question is how does the universal gravitational constant, G, which determines how strongly gravitating masses attract each other, know what value to assume if there is no other mass in the universe. Mach proposed, essentially on philosophical grounds, that G must be determined by the sum total of all of the mass in the universe. Einstein assumed in his General Theory of Relativity that G is simply a universal constant, independent of the specific mass distribution of the universe. On the other hand, Brans and Dicke later proposed a so-called scalar-tensor theory of gravity in which the local value of G depends upon the rest of the mass in the universe through an additional scalar field that does not appear in Einstein's theory. (special thanks to Warren Davis, Ph.D., President, Davis Associates, Inc., Newton, MA USA for this answder)