If I am travelling on a car at around 60 km/h, and I shine a light, does that mean that the light is travelling faster than the speed of light? The title says it all.
If I was on a bus at 60 km/h, and I started walking on the bus at a steady pace of 5 km/h, then I'd technically be moving at 65 km/h, right?
So my son posed me an interesting question today: since light travels as fast as anything can go, what if I shined light when moving in a car?
How should I answer his question?
 A: 
If I was on a bus at 60 km/h, and I started walking on the bus at a
  steady pace of 5 km/h, then I'd technically be moving at 65 km/h,
  right?

Not exactly right.  You would be correct if the Galilean transformation correctly described the relationship between moving frames of reference but, it doesn't.
Instead, the empirical evidence is that the Lorentz transformation must be used and, by that transformation, your speed with respect to the ground would be slightly less than 65 km/h.  According to the Lorentz velocity addition formula, your speed with respect to the ground is given by:
$$\dfrac{60 + 5}{1 + \dfrac{60 \cdot 5}{c^2}} = \dfrac{65}{1 + 3.333 \cdot 10^{-15}} \text{km}/\text h \approx 64.9999999999998\ \text{km}/\text h$$
Sure, that's only very slightly less than 65 km/h but this is important to your main question because, when we calculate the speed of the light relative to the ground we get:
$$\dfrac{60 + c}{1 + \dfrac{60 \cdot c}{c^2}} = c$$
The speed of light, relative to the ground remains c!
A: You can start in answering his question by explaining the Doppler shift for acoustical waves.

The Doppler effect (or Doppler shift), named after the Austrian physicist Christian Doppler, who proposed it in 1842 in Prague, is the change in frequency of a wave (or other periodic event) for an observer moving relative to its source. It is commonly heard when a vehicle sounding a siren or horn approaches, passes, and recedes from an observer. The received frequency is higher (compared to the emitted frequency) during the approach, it is identical at the instant of passing by, and it is lower during the recession.
The relative changes in frequency can be explained as follows. When the source of the waves is moving toward the observer, each successive wave crest is emitted from a position closer to the observer than the previous wave. Therefore each wave takes slightly less time to reach the observer than the previous wave. Therefore the time between the arrival of successive wave crests at the observer is reduced, causing an increase in the frequency. While they are travelling, the distance between successive wave fronts is reduced; so the waves "bunch together". Conversely, if the source of waves is moving away from the observer, each wave is emitted from a position farther from the observer than the previous wave, so the arrival time between successive waves is increased, reducing the frequency. The distance between successive wave fronts is increased, so the waves "spread out".

Your son's expectation works on this intuitive background.
But light waves, in contrast to sound waves which need air to reach our ears, do not need a medium to reach our eyes. This is evident in that the light from stars reaches us through the vacuum of space where there is no medium. People used to hypothesize a medium for light, aether but experiments proved, as the other answers state correctly, that the velocity of light was constant, c,  no matter what the motion of the emitter  or absorber. Thus no, there will be no change in the velocity measured of the emitted light whether we are sitting on the ground, forward or backward or sideways, or in the car itself.
There is an effect though. Light that has been emitted by a source moving towards us does not change its velocity but it does change its frequency to a higher value; if it is receding, to a lower value. As the energy of the photons is given by E=h*nu it means that it gains an extra energy or loses some due to the relative motions of observer and emitter.
This has been very useful for astrophysics. For example that is how we know the relative motions of stars with respect to us. Light comes from spectra of atoms and we know them here in the lab. They are distinctive and identify whether we see light from iron or oxygen or hydrogen in a gas state. The change in frequency of the spectral lines will tell us of the motion of the star relative to us. There exist many applications of this method.
A: As a matter of technicality, I believe that shining from a moving vehicle on earth will in fact be faster* than the light shining from a stationary vehicle on earth, however both forms of light would be moving slower than the speed of light (c), which is referencing the speed of light in a vacuum.
This is because the medium that the light is moving through (air) slows down the light by about 88km/s (according to Wikipedia).
That said, light in a vacuum emitted from a moving object should travel at the same speed as light in a vacuum emitted from a stationary object for the reasons outlined by all the other answers to this question.

* so long as the light is propagating in air that is traveling with the vehicle, such as the air between the headlight bulb and the headlight casing
A: Second postulate (invariance of c) of the special theory of relativity goes like this:

As measured in any inertial frame of reference, light is always propagated in empty space with a definite velocity c that is independent of the state of motion of the emitting body.

Or, that objects travelling at speed c in one reference frame will necessarily travel at speed c in all reference frames. This postulate is a subset of the postulates that underlie Maxwell's equations in the interpretation given to them in the context of special relativity.
So basically, there exists an absolute constant 0 < c < (infinity) with the above property. So you can shine light while travelling at the speed of light and it will still go at c, not more, not less.
Ref: wiki.
A: One essential postulate of special relativity is that light moves at the same velocity in all reference frames. Somebody standing next to the moving bus will observe the light travelling just as fast somebody who is on the bus sees it. It might not be intuitive, but it is consistent both with experiment and the mathematical framework of the theory. 
A: You should tell your son that this very question was asked by, explored by, and eventually answered by the some of the brightest physicists of the 19th century. Eventually two scientists named Michelson and Morley came up with an experiment to measure this effect, and were amazed to discover that it didn't exist! Rather:
Light travelled at exactly the same speed in all directions, regardless of any velocity of its emitter.
This result astounded the physicists of the world, and led to the development of the Special Theory of Relativity by Einstein.
A: Your son is correct, from the perspective of an outside observer who is not on the bus. The same is correct with the light. While 65km/h is negligible in relation to the speed of light, he is right. If I am traveling at 90% c and I shine a light ahead of me, the light will leave me at the speed of light. If you, standing still, observed me, the light would still leave my flashlight at c. Weird right, something has to give to balance this out, and this is time. Which means in my example, someone is time traveling. Remember that everything you see with your eyes, you are seeing it in the past, not now at the exact moment. Einstein realized this when he would ride the train and look at the clock on a building.
A: Ok, so that's a classic question that depicts how wrong we think of reality, or maybe how much diffrent reality is from how we think of it.

Every object moves in spacetime at one and only one speed. The speed of C=1ls/s (aka: one lightsecond per second, also known as "the speed of light" but in reality its the speed of everything that moves in spacetime).

The only difference between light and say a car... Is that light moves at the speed of C solely in space. Where a car moves at the speed of C in both space and time combined.
That combination of speeds (your speed in time and in space) is called your speed in spacetime and its related to the Lorentz factor γ.
$$γ\ =\ \frac{1}{\sqrt{1-\frac{u^{2}}{c^{2}}}}$$
where:

*

*v is the relative velocity between inertial reference frames

*c is the speed of light in a vacuum

*β is the ratio of v to c

*t is coordinate time

*τ is the proper time for an observer (measuring time intervals in the observer's own frame)


Now... Lets take a look at a photon and a car.
A car is moving at 60km/h (or 16.6 m/s) in space and using the Lorentz factor we can see that the car is ALSO moving at (299,247,994.113 m/s) in time. If we combine 16.6 and 299,247,994.113 for spacetime, we get 299,792,458 m/s which is exactly, the speed of C (or the "speed of light") in spacetime. So your car is moving at the speed of light in spacetime.
Now a photon is even easier!
A photon is moving at 299.792.458 m/s in space. And using the Lorentz factor we can find that the photon is moving at exactly 0 m/s in time!
So again, if we combine 299.792.458 and 0 for spacetime we will get 299.792.458 m/s which is the speed that the photon is traveling in spacetime.
Both the photon and the car are traveling at the same exact speed, in spacetime! The speed of C. The car is mostly moving in time, the photon is ONLY moving in space, that's the only difference between them.

So, your question was "If I am travelling on a car at around 60 km/h, and I shine a light, does that mean that the light is travelling faster than the speed of light?"
Lets rephrase that question to make it more accurate:
"If I am traveling on a car at around 60km/h IN SPACE (and 299,247,994.113 m/s in time), and I shine a light, does that mean that the light is traveling faster than the speed of light in space?"
NO
Your car will be moving at the speed of light but mostly moving in time and a little bit (16.6 m/s) in space. The photons from your flashlight will be moving at exactly the speed of light in spacetime but ONLY in space and not in time. Both you and the photons will continue to move in spacetime at exactly the speed of light, your separate trajectories though won't be the same. You will continue to travel mostly in time and the photons will travel only in space and time for them won't be passing.
That's the real reality. The reality that we observe is false, because we don't see (or can't see) the real picture. We can't see that the spacetime around us is 4 dimensional and hyperbolic. Also we forget that things also move in time and not just in space. That's why we see all those effects of relativity (like time dilation and lenth contraction). If we could see spacetime from a bird's eye view, we would see that the only thing that you can really do is change your "trajectory" in spacetime (performing a hyperbolic rotation). You can't really accelerate in spacetime because the more you speed up in space the more you slow down in time, the net result will continue to be the speed of C everytime!
