Bliss Sloan Answers Readers: Questions on the Speed of Light

From Bliss Sloan's answers to questions from readers on AOL...

[EagleRidge Home] | [Resources] | [Sloan] | Speed of Light from a Pick-up Truck's Headlights

Question:

"I've convinced everyone here that light, at 186,000 miles per second, is the fastest thing on earth. Melba, sitting in her pick-up truck, turns her headlights on and that light strikes the garage doors at 186,000 mps....imagine that? Melba wants to know: When she's toolin' down Main street at 60 m.p.h. and turns her headlights on....how fast is that light traveling? ...then adds, when those astronauts shine their flashlights out the front of the spacecraft, how fast is that light traveling? Melba thinks she may have discovered a new speed record. Whadda ya think?"

Answer:

Good questions!

First of all, contrary to popular belief, light is NOT the "fastest thing on earth," just the "fastest signal in a vacuum." On earth, light travels through the air at a speed slightly less than c (c=299,792,458 meters per second), a.k.a. the 186,000 miles per second speed you mention. Light only travels at c in a vacuum. For example, some other signals, such as high-speed cosmic rays, can travel faster than light through water. The speed of light in a medium is:

c_in_medium = c / n

where n is the refractive index of the medium and is always greater than 1, making the speed of light in a medium always less than c.

See "Light in Moving Media" for further discussion of this topic:
[http://www.st-andrews.ac.uk/~ulf/media.html]

That said, I'll go back to the question at hand. Einstein's famous Theory of Relativity had two parts:

that the laws of space-time should be the same from all reference frames (i.e., points of view);
and
A postulate: the speed of light in a vacuum is the same for all reference frames.

In other words, the Theory of Relativity assumes that the speed of light in a vacuum is a constant. However, this assumption is based on solid experimental evidence, including the Michelson-Morley experiment (see:
[http://scienceworld.wolfram.com/physics/Michelson-MorleyExperiment.html]),
and the Fizeau experiment (see:
[http://scienceworld.wolfram.com/physics/FizeauWheel.html]).
Evidence so far says that this assumption of the constancy of c is accurate to at least 1 part in a billion. Of course the constancy of c is still only a theory or postulate, not proven fact. But the facts so far indicate that c is indeed constant.

Now, to approach your question in more detail, supposing for the moment that the pick-up truck, headlights, and garage door are in a vacuum, and that your friend Melba is wearing a space suit for the occasion:

If we just add the two speeds in your example (60 miles per hour + 186,000 miles per second), then the resulting speed of the light from the headlights would be larger than c. However, the above experiments show that this is not the case. We cannot just add two speeds in the same direction to get the resulting speed! Though adding the two speeds seems OK to us for relatively slow speeds (much slower than c), even for those cases plain addition is not quite accurate. Of course, for most purposes ("If Train A is heading south at 30 mph, and the engineer shoots an arrow south from the front of the train..." ) plain addition works well enough, since the relativistic correction would be very small.

What to do? If we cannot just add two speeds (such as adding the speed of the car to the speed of the light from the headlights) to get the resulting actual speed, how DO we calculate the effect of adding one speed on top of another?

Here it is:

v_result = (v1 + v2) / ( 1 + ((v1*v2)/(c*c)))

That is, the resulting speed is equal to the familiar sum of the original speeds (v1 and v2), DIVIDED BY 1 plus (v1*v2 divided by the square of c).

You can see the effect of different speeds by calculating the above formula for different speeds v1 and v2. In your example, where v1 = 60 mph = 1/60 miles per second, and v2 = c = 186,000 miles per second:

                  (1/60  +  186,000) miles per second 
v_result =   ------------------------------------------------
                  1 +     (1/60 * 186,000) 
                           ------------------------- 
                         (186,000 * 186,000)


                      (1/60 + 186,000) miles per second
                  =    -------------------------------------------
                       1 +        1/60
                                 ------------------
                                 186,000

Continuing to simplify:

                    
                   1/60 + 186,000 miles per second
v_result =    --------------------------------------------
                        186,000     +     1/60
                        ----------------------------
                                   186,000

                        1 mps
             =    -------------------
                          1
                      ------------
                      186,000

             =  186,000 miles per second = c

There *is* a "headlight" effect that happens when a headlight is moving (see:
[http://lightspeed.sourceforge.net/])
but making the light go faster than c is not part of it.

Some interesting links:

Adding Velocities: A Walk on the Train
[http://galileo.phys.virginia.edu/classes/109N/lectures/adding_vels.html]
Leftover Valentine's Chocolate: Use It to Measure the Speed of Light - excellent idea for a home science experiment from the February, 2010, issue of Wired Magazine.
The Speed of Light - A Limit on Principle? A physicist's view on an old controversy:
[http://home.sunrise.ch/schatzer/space-time.html]