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==A Topical Example: Speed of Extremely Energetic Neutrinos==

==A Topical Example: Speed of Extremely Energetic Neutrinos==

Here is another example of the use of this formula in physics calculations.  Recently there has been quite a controversy over whether neutrinos were observed traveling at a speed faster than light.  Relativity doesn't allow that, and, since neutrinos have nonzero (but incredibly tiny) mass, they aren't even supposed to travel ''at'' the speed of light.  

Here is another example of the use of this formula in physics calculations.  In 2011 there were [http://www.theguardian.com/science/2011/sep/22/faster-than-light-particles-neutrinos?newsfeed=true reports] that high-energy neutrinos had been observed traveling at a speed faster than the speed of light in an experiment at the Gran Sasso laboratory in Italy.  Specifically, they seemed to have arrived at the detector 60 nanoseconds faster than light would have.  Relativity doesn't allow that, and, since neutrinos have nonzero (but incredibly tiny) mass, they aren't even supposed to travel ''at'' the speed of light.  

The mass of a neutrino is about 0.44x10^-36kilograms.  (Normally all of these things are measured in more convenient units such as Giga-electron-Volts, but that makes implicit use of E=mc².  If we don't accept that, we have to do the calculations under classical physics, using SI (meter/kilogram/second) units.)  The neutrinos were accelerated to an energy of about 17GeV, or .27x10^-8Joules.  Using the classical formula <math>\mathrm{E} = \frac{1}{2}mv^2</math>, we get v=110x10¹²meters per second.  This is about 370,000 times the speed of light.

The mass of a neutrino is about 0.44x10^-36kilograms.  (Normally all of these things are measured in more convenient units such as Giga-electron-Volts, but that makes implicit use of E=mc².  If we don't accept that, we have to do the calculations under classical physics, using SI (meter/kilogram/second) units.)  The neutrinos were accelerated to an energy of about 17GeV, or .27x10^-8Joules.  If one did not accept relativity and had to use classical physics and the classical formula <math>\mathrm{E} = \frac{1}{2}mv^2</math>, one would get v=110x10¹² meters per second.  This is about 370,000 times the speed of light, something that scientists would certainly have noticed. In fact, with special relativity, the speed is just under the speed of light, such that the neutrinos should be received at the detector about .26x10^-24 seconds (.26 yoctoseconds) later than the speed of light itself.  This is far too small to measure&mdash;15 orders of magnitude smaller than the resolution of the GPS signals in the experiment.

However, the classical formula breaks down at speeds close to <math>c</math>, and indeed, as the speed of a massive object approaches <math>c</math>, the object's kinetic energy approaches <math>+\infty</math>.

Several scientists have gone on record stating that the neutrinos, which have mass, travel at precisely the speed of light.  If true, this disproves the [[Theory of Relativity]] and the claim that '''E=mc²'''. However, it is more likely that those scientists are using language inaccurately. It is impossible to measure the speed of neutrinos precisely. What is meant is the difference between the speed of light and the speed of the neutrinos is too small to measure.

Later [http://news.sciencemag.org/2012/02/official-word-superluminal-neutrinos-leaves-warp-drive-fans-shred-hope%E2%80%94barely?ref=hp reports] started to resolve the mystery, and it is now accepted that the neutrinos behaved properly.  But a BBC reporter made the incorrect statement that [http://www.bbc.co.uk/news/science-environment-17364682 the neutrinos travelled at precisely light speed].  This was a simple misstatement, by .26 yoctoseconds.

 

The issue was discussed at length at Conservapedia.<ref>http://www.conservapedia.com/Talk:Main_Page/Archive_index/102#Faster_than_light_neutrinos</ref> <ref>http://www.conservapedia.com/Talk:Main_Page/Archive_index/102#The_final_nail_in_the_coffin_of_relativity.3F</ref> <ref>http://www.conservapedia.com/Talk:Main_Page/Archive_index/102#Another_Blow_to_Relativity</ref> <ref>http://www.conservapedia.com/Talk:Main_Page/Archive_index/109#Neutrinos_now_obey_speed_limit</ref> <ref>http://www.conservapedia.com/Talk:Main_Page/Archive_index/111#Neutrinos</ref>

==Deducing the Equation From Empirical Observation==

==Deducing the Equation From Empirical Observation==