Bingo!
Here is an easy one.
"Ok, three things must ALWAYS be born in mind when talking about this sort of thing.
1. Nothing can go faster than light speed
2. Light always moves at lightspeed
3. Speed is relative.
The first is a given
The second is important to bear in mind- unless acted upon by some force, light always goes at lightspeed. You don't get slow light. You have a lightbeam, lightspeed is what it wants to go at
The third is vital. For example: The Earth is meant to move at about some 10,000 miles an hour. But that is only relative to the sun. To US the Earth does not move at all. Who is right? Both! There is no central point that we measure speed from, no 'zero' speed area. ALL speed is relative. Relative to the centre of the galaxy, the sun and all the planets are moving at huge spees. Relative to us that movement does not exist. Driving on a motorway, the people in the car with you are not moving at all. Relative to the people coming the other way in cars, they are moving at nearly 100mph or more. BOTH are correct; it's all relative.
Ok, so, once that is bourne in mind... here are some examples about speed that help answer this.
Imagine you are on a bike chasing a car. The car will get away from you, in all probability. Logically, though, the harder you pedal, the faster you will go, and the less quickly the car will escape you. Fair enough, right?
Now, imagine you are not chasing a car but a beam of light. Remember that the speed is always relative to you, and the light beam always moves at lightspeed. This means that no matter HOW fast you pedal, the lightbeam will always get away from you just as quickly! Is the lightbeam speeding up to do this? No! It's just moving away from you at the same relative speed- lightspeed!
Hang on. What about your friend standing at the starting line? How does that work for him? If the lightbeam is getting away from you, on the bike, at lightspeed, then surely it will look to him as if it is moving at MORE than lightspeed! Well, actually, no. To him it will look like it is moving at lightspeed from him. The rules still apply.
Hang on. Doesn;t that mean that the light beam is moving at different speeds for each of you? Well, no. For both of you it is moving at lightspeed. But what WILL happen is this- the guy on the bike will see the lightbeam hit and illuminate objects BEFORE the person on the starting line does, by a fraction.
Doesn't that defy common sense? Well, it seems to. One of the basic tenets of relativity is that two people will see the same thing in different ways. The importsnt thing is that the same events still happen, the lightbeam IS there, but you don't see it all happening at the same time.
So, to return to the question... we need two people to answer it. One in the car, and one watching the car drive by.
For the person in the car, nothing odd happens. The headlights are not moving, relative to him! So when he turns them on, the lightbeam will happily move away from him at lightspeed just as it would on a motorway.
But for the person watching it drive by.... the car (and so the headlights) are moving at lightspeed. So where does the light go? It goes nowhere. With the car moving at lightspeed already the lightbeam cannot improve on it. Therefore this person will see no light coming out the headlights.
Which returns to my basic point, what happens depends where you are in relation to the light source!
Mad, isn't it?
All of this, of course, ignores the fact that you can neither move at lightspeed nor observe any object doing so!"