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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?

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    $\begingroup$ If you like this question you may also enjoy reading this Phys.SE post. $\endgroup$ – Qmechanic♦ Oct 2 '13 at 12:18
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    $\begingroup$ This is the amazing thing about relatvivity that the velocity of light does not add the same way as small velocities do! Your son is on the path which led Einstein to develop relativity theory. $\endgroup$ – Slaviks Oct 2 '13 at 13:10
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    $\begingroup$ This should answer your question in less than five minutes: youtube.com/watch?v=vVKFBaaL4uM $\endgroup$ – shortstheory Oct 2 '13 at 17:47
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    $\begingroup$ This is exactly the question Einstein asked, which led him to discover special relativity. $\endgroup$ – BlueRaja - Danny Pflughoeft Oct 2 '13 at 18:26
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    $\begingroup$ Nobody mentioned Michelson-Morley experiment yet - amazing, as that is the empirical basis of everything below, and is simple enough for a child to understand the results (if not the techniques) of: en.wikipedia.org/wiki/Michelson%E2%80%93Morley_experiment $\endgroup$ – Pieter Geerkens Oct 2 '13 at 22:11
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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!

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    $\begingroup$ True, but usually we think of the fact that the speed of light is constant as being a law of physics, and the Lorentz transformation is just a consequence of that law. $\endgroup$ – 200_success Oct 2 '13 at 22:06
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    $\begingroup$ A correct answers, but hardly comprehensible to a 6 or 8 year old. $\endgroup$ – Pieter Geerkens Oct 2 '13 at 22:19
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    $\begingroup$ @PieterGeerkens No, but the idea of Alfred's answer is some thing children of that age love to hear - I think it would be a grand and appropriate thing to tell them that "it's actually a teeny bit less that 65kmh, and the faster you go, the more the rule that speeds add together in the way that children understand is broken". $\endgroup$ – Selene Routley Oct 2 '13 at 22:57
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    $\begingroup$ @WetSavannaAnimalakaRodVance: You honestly believe that bright children like being talked down to like that? Shame on you! $\endgroup$ – Pieter Geerkens Oct 3 '13 at 2:46
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    $\begingroup$ @PieterGeerkens In some cases yes: children are all very different. Education research shows that this age is the very age where a system's abstract properties are only just beginning to take root in their minds - see here - so algebraic explanations befuddle many children. It's important to know your student well and watch their reaction to you carefully - clearly geek-girl or boy is not going to forgive you for a too simplistic answer so you need to be ready to shift gears in a split second. But, given what we know about childrens ... $\endgroup$ – Selene Routley Oct 3 '13 at 4:18
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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.

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    $\begingroup$ +1 Great answer. I love how it touches on history of science, the empirical method, and the story of this discovery. Sure to lead to OP's son wanting to find out more. $\endgroup$ – Ergwun Oct 3 '13 at 2:34
  • $\begingroup$ They were actually checking the velocity of it's emitter respect to the ether, which they wanted to prove. Only to find that there was no ether. $\endgroup$ – Francisco Presencia Oct 3 '13 at 10:40
  • $\begingroup$ @FrankPresenciaFandos: Yes, by measuring the increased speed of light in the direction in which the Earth was moving relative to the aether. $\endgroup$ – Pieter Geerkens Oct 3 '13 at 11:10
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    $\begingroup$ "this very question was asked by, explored by, and eventually answered by the some of the brightest physicists of the 19th century" In other words, in asking this very question, OP's son was in some very good company! $\endgroup$ – user Jun 30 '16 at 15:22
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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

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    $\begingroup$ The first paragraph is wrong. Even in media where the speed of light is less than $c$, the speed of propagation is still independent of the speed of the source. $\endgroup$ – user10851 Oct 2 '13 at 23:22
  • $\begingroup$ @ChrisWhite, I'd be more than happy to update my answer if you'd elaborate on why that's the case. My knowledge of physics is superficial at best, so I'd much rather have more good information than lead people astray if I'm wrong. $\endgroup$ – zzzzBov Oct 3 '13 at 2:53
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    $\begingroup$ Waves - whether they are sound or light or water - are local phenomena, which means how the wave decides to move is determined by its immediate surroundings, not by whatever the source may have been doing far away. The speed of the source affects the frequency of the wave, but two waves of the same frequency in the same medium will move at the same speed, which for light in air is, as you said, a bit slower than $c$. $\endgroup$ – user10851 Oct 3 '13 at 3:04
  • $\begingroup$ @ChrisWhite, then in that case: if a two light waves were traveling in two separate media of the same substance in the same direction, and those media were moving in relation to each other in that direction, wouldn't the light waves be traveling at different speeds with regards to each other? $\endgroup$ – zzzzBov Oct 3 '13 at 4:34
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    $\begingroup$ If the air was moving with the car, the speed of light could be faster that the speed of light through still air. But if it is only the car that is moving and not the air, then both stationary and moving flashlights will put out light of the same speed, ie c minus 88 km/s. $\endgroup$ – Mark Lakata Oct 3 '13 at 5:14
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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.

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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.

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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.

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    $\begingroup$ Should read "... in all inertial reference frames.". $\endgroup$ – Pieter Geerkens Jun 30 '16 at 15:38

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