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Muon Decay: CR4 Challenge (05/12/08)

Posted May 11, 2008 5:01 PM
Pathfinder Tags: challenge questions

This week's CR4 Challenge Question:

John and Tom were fishing when John related how he had recently read that Muons had a lifetime of only 2.2 microseconds and 10,000 Muons per square meter per minute hit the Earths surface. The Muons are created when high energy protons collide with molecules 100 km up in the atmosphere. Tom did some quick math in his head and said "If the Muons traveled at the speed of light, they could only travel about 650 meters before decaying, how do they make it the 100 km to the Earth's surface?" What was John's answer?

(Update: May 20, 8:57 AM EST) And the Answer is...

John explains that the Muons are traveling near the speed of light and therefore experience time dilation, which allows them to travel the 100 km to the Earth's surface.

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#1

Re: Muon Decay: CR4 Challenge (05/12/08)

05/11/2008 6:50 PM

Not all the muons decay after 2.2 microseconds. Only half of them do, hence the term "half life". In one half life they travel ~660m, therefore ~152 half lives elapse during their 100km trip through the atmosphere. So if 10,000 muons/m2 hit the earth each second then they must be 'generated' at 2151 times this rate.

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#2
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Re: Muon Decay: CR4 Challenge (05/12/08)

05/11/2008 7:58 PM

I think the other half of this is that not all muons are created 100 km from earth. Some of the high energy protons reach closer to earth before impacting an atom and creating a muon.

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#3
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Re: Muon Decay: CR4 Challenge (05/12/08)

05/11/2008 10:35 PM

Come to think of it I don't know enough much muons. When they are created up there do they all head towards earth or are they equally likely to head in any other direction?

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#4

Re: Muon Decay: CR4 Challenge (05/12/08)

05/12/2008 2:07 AM

If the Muons had a lifetime (and not a half life) of 2.2 microseconds then this is how long it would take them to decay in their own inertial frame of reference, however, as they will be travelling at some speed relative to the earth their internal 'clocks' will be slower when measured by someone in the frame of reference of the earth. This is due to Einsteins special theory of relativity which shows that observers in one (non accelerating) reference frame see time moving slower in another (non accelerating) reference frame which is moving at constant velocity relative to them by a factor of 1/(1-B2)1/2, where B = v/c, (v = relative velocity, c= speed of light) . For the muons to travel the 100km to the earth without decaying they would need to have their internal 'clock' slowed by a factor of approx 151 (as measured by observer on earth) which means they would have to be travelling at a velocity of about .99998 of the speed of light.

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#7
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Re: Muon Decay: CR4 Challenge (05/12/08)

05/12/2008 8:09 AM

You are quite correct Bob, see info below C&P from wiki;

On earth, all naturally occurring muons are apparently created by cosmic rays, which consist mostly of protons, many arriving from deep space at very high energy.

About 10,000 muons reach every square meter of the earth's surface a minute; these charged particles form as by-products of cosmic rays colliding with molecules in the upper atmosphere. Traveling at relativistic speeds, muons can penetrate tens of meters into rocks and other matter before attenuating as a result of absorption or deflection by other atoms.

—Mark Wolvertron, science writer, Scientific American magazine, September 2007, page 26 "Muons for Peace"

When a cosmic ray proton impacts atomic nuclei of air atoms in the upper atmosphere, pions are created. These decay within a relatively short distance (meters) into muons (the pion's preferred decay product), and neutrinos. The muons from these high energy cosmic rays, generally continuing essentially in the same direction as the original proton, do so at very high velocities. Although their lifetime without relativistic effects would allow a half-survival distance of only about 0.66 km at most, the time dilation effect of special relativity allows cosmic ray secondary muons to survive the flight to the earth's surface. Indeed, since muons are unusually penetrative of ordinary matter, like neutrinos, they are also detectable deep underground and underwater, where they form a major part of the natural background ionizing radiation. Like cosmic rays, as noted, this secondary muon radiation is also directional. See the illustration above of the moon's cosmic ray shadow, detected when 700 m of soil and rock filters secondary radiation, but allows enough muons to form a crude image of the moon, in a directional detector.

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#26
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Re: Muon Decay: CR4 Challenge (05/12/08)

05/17/2008 12:38 AM

Hello design-engineer:,

this is a very interesting post.

Are the muon's one of the particles which pass straight through our bodies?

And is it this that ages everything (partly) by damaging the three or four human cells they come in contact with and when hitting them 'injure' them and start breaking them down............Hence the aging?

jfmfit

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#28
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Re: Muon Decay: CR4 Challenge (05/12/08)

05/17/2008 6:49 AM

Hi, jfmfit!

It's difficult to imagine that particles passing through the human body do not cause damage to the cellular structure in some sense, and might even contribute to aging. However, there are 10,000 such particles passing through a very small space every second, and in an accumulated amount of time, millions and millions of them. Muons are only one of proabably dozens of these. With that many little bombs hitting us all the time, we must have developed some sort of immunity against them in order to remain alive. What I'm expressing is the thought that they don't do much damage, if any, to any species that has evolved under their influence as they go through.

The human body has evolved within this milieu. Cell membranes are permeable, allowing things to 'pass through' them, and closing up right afterward. But these particles are so small that they pass between the atoms that make up everything, and so numerous that our poor body cells would never be able to completely close around the 'openings' that they make, should they actually make any.

Our brains work with minute chemically-induced electrical signals. Imagine what kind of interference the particulate bombardment would impose on that, if it were an effective form of interference.

It's been said that 'cosmic rays' contribute to aging. Certainly, over-exposure to the sun has been said to promote premature aging of the skin due to UV presence.

You could be correct. Interesting question.

Mark

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#30
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Re: Muon Decay: CR4 Challenge (05/12/08)

05/17/2008 7:41 PM

Hello once again Mark The Handyman,

I just love that word 'milieu'. I think where you find a word as good as that it should be fitted into a sentence as often as possible!

I understand all you say...............(surprisingly for me!) I have a book called 'A sea of oxygen' or something like that. Though I do not have it to hand at the moment, it explains Oxygen is the main cause of aging. As it does with metal rusting, the skin and all cells literally 'oxygenizes' and it leaves tiny scars which eventually over time make, not only the skin but every cell in our bodies because they are all open to the need and ravages of oxygen, age by the damage they do over our lifetime.

I don't know where you are but here in the UK there is some new cream out "infused with oxygen", which is rather odd, as the more cream applied ages the skin more, and not as said, make you look younger!

jfmfit

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#5

Re: Muon Decay: CR4 Challenge (05/12/08)

05/12/2008 7:56 AM

It is possible, when discussing nuclear scale particals, that a lot of these protons are not colliding with molecules until they get closer to the earth. The protons can actually travel between the molecules, since they are smaller and will likely even travel through the electron cloud of the atoms in the molecules due to their small cross sectional area. With this, the collisions can be a lot closer to the earth's surface than originally thought.

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#6
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Re: Muon Decay: CR4 Challenge (05/12/08)

05/12/2008 8:08 AM

Hey, I looked up the muon on Wikipedia and it looks like there are a number of reasons for the difference. Time dilation is part of it and also the muons can travel through dense matter, such as rock. See link:

http://en.wikipedia.org/wiki/Muon

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#10
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Re: Muon Decay: CR4 Challenge (05/12/08)

05/12/2008 11:18 PM

Wasn't a Muon detector tank built deep underground and the were detecting not just from above, but upwards thru the ground as well?

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#8

Re: Muon Decay: CR4 Challenge (05/12/08)

05/12/2008 10:51 PM

Isn't a Muon a type of sheep?

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#9
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Re: Muon Decay: CR4 Challenge (05/12/08)

05/12/2008 11:09 PM

Is that a baaa-ad joke, or what?

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#14
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Re: Muon Decay: CR4 Challenge (05/12/08)

05/13/2008 5:27 AM

Joke? What joke?

Sorry just trying to pull the wool over your eyes.

Boom Boom, ta-ching.

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#11

Re: Muon Decay: CR4 Challenge (05/12/08)

05/12/2008 11:59 PM

Muons were traveling at or near the speed of light. Therefore, they were in a different time "dimension" and could travel that distance in their own time while we observe from our time dimension and wonder "Whaaa?" But that would almost mean two different time dimensions could exist at the same - what should it be called, the same time?

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#12
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Re: Muon Decay: CR4 Challenge (05/12/08)

05/13/2008 2:27 AM

Special relativity is one of the most counter-intuitive theories in physics and yet it has been verified to an extreme level of accuracy in numerous experiments. It applies to any reference frames that have any relative motion not just when the relative velocity approaches the speed of light. It is only at the higher velocities, however, that the effect is significant enough to be measured. There is no new time dimension that the muons enter when they approach the speed of light. The dimension of time (like space) is relative depending on the velocity between the observer and the object being observed. If the muon was sitting on a train moving at 50km/hr past the platform it would still decay slower (relative to someone on the platform) than if it was sitting on the platform next to the observer, however, at that velocity the difference would be so small that it would not be able to be measured.

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#13
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Re: Muon Decay: CR4 Challenge (05/12/08)

05/13/2008 3:18 AM

Could the lifetime of a Muon be much longer than stated, cause the observed life time of it, is the time it takes between winking into then back out of out plane of existance, as it traverses the parallel worlds?

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#15

Re: Muon Decay: CR4 Challenge (05/12/08)

05/13/2008 6:51 AM

So muons are still making it the news after all this time, are they? According to the Wickipedia article on the little buggers, they really whistle along...to the point where they are part of cosmic rays when they enter the earth's influence, and travelling at "relativistic speeds" (not to be confused with the amount of time the wife takes to get ready...this is another relative!).

When you're in that much of a hurry, you can cover a lot of territory in a very short time, even when you lose half your life in a mere blink of an eye. Hence, the muons make it all the way into the underworld as they haven't even got time to stop at the surface to say a decent "hello! glad to be here!"; and then, without a word of explanation, disappear...which is an interesting phenomenon within itself, because even burned out things don't really just 'disappear'.

Since they are similar to big fat electrons and too rude to even say hello, they were originally going to be called "big fat rudons". The Ferengi scientist who discovered them, however, had a mother-in-law (oh no! not another mother-in-law joke!) who, you guessed it, also fitted this dubious description, and so he named it after her; and hence we are saddled forever with muons.

Ah well, at least they're in a hurry. And we know they're making it into the underworld!

John, who had a mother-in-law and knew the answer, didn't say anything. He just smiled, and to the casual observer rather cruelly bated the next worm onto his current hook.

Mark

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#21
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Re: Muon Decay: CR4 Challenge (05/12/08)

05/14/2008 11:44 PM

Actually, the scientist that "discovered" muons was a alien. He's from Felinios, the planet of the cat-people. They have used "mew-ons" for centuries as an energy source. He came to this planet as part of a plot to slowly introduce advanced technology into our culture to make us easier to assimilate. Naming the particle after the Greek letter μ is just a convenient cover story.

Think about it. Del the Cat is one of the more active members of CR4. I have noticed several other members have cat avatars. And Hendrik? A lion is a big cat after all. Then there's that "Engineers' Guide to Cats" video.

Also, there's that joke about dogs having owners while cats have staff.

Need any more evidence?

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#22
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Re: Muon Decay: CR4 Challenge (05/12/08)

05/15/2008 7:09 AM

Hi, 3Doug!

Well, this would indeed seem to pose a conundrum , for as you have noted, mewonic energy has been around for ages.

[As a side note, however, we don't have to worry about cats taking over. They are just pets, after all, and we feed them and put them down when they get old. Also, as is evidenced by Del and others, including personal acquaintances of mine like Minerva, Caesare, Zoe and Scottie, their stealth is highly over-rated. Just observe, despite the obvious intelligence of his contributions, the lack of subtlety of Del's claims to want to take over the world so he can be the center of attention in his own blogging. He just wants to be a pet cat, only Del's desire to take over and be thus-served is megalomaniacal instead of just familial. (And, in spite of its obvious advantages and advanced technology, you may have heard of mewonic energy referred-to jokingly as "me-want-it now energy" ...a gentle poke at the cat people's mentality around attention-seeking.)]

Mewonic energy has been around for a long time. But then, so has moo-onic energy, known here on earth as methane; but having been (according to recorded history) first observed and utilized on either KineyWine, the planet of the cow people (and let's not go into that sorrowful aspect of human history...we still eat them!), or by other ancient civilizations like the various Termitic cultures of the universe.

I think if you can get access to a copy of Purr-posepedia from Felinios (not exactly sure if that's the title, although it's something like that, I think); or the Ferengi Energy Acquisition Rule Book with the historical appendices, you will note that mewons --although of an apparently random scooting nature, and making a distinct squeaking noise-- are a more powerful, manageable and predictable energy source than muons, which are silent, ephemeral and fleeting in nature. The Ferengi physicists joke about 'silent muons'. (Because of the "mu" mother-in law connotation, it's a part-oxymoron.) That's also where the ironic expression "silent as a muon", which is rapidly gaining popular usage throughout the known universe, originated.

Both are sub-atomic particles, but the muons have to originate with nuclear energy burst activity and are a second-hand observation of the same, whereas the mewons are the activity, if you see what I mean. If it were possible to use muons as an energy source, you would first have to build an energy source to make them. And as you well know, the Ferengi Energy Acquisition Book rule number 14 (or was it 15?) is "Any utilization of energy must efficiently be able to create a profit"...their nod to energy efficiency.

Mark

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#23
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Re: Muon Decay: CR4 Challenge (05/12/08)

05/15/2008 1:52 PM

Mark, your reply shows how deceptive their plan is. Cats are not seeking to take over in the sense of attacking and conquering as humans would. They seek to make us willingly surrender to their furry cuteness and cuddliness. Obviously, they have gotten to you.

Of course all cats and cat-people wish to be the center of attention. Del is not a leader in their efforts. He is just a follower, or a foot soldier (maybe that should be paw-soldier). By drawing our attention, he makes it easy for us to succumb to their charms.

The Ferengi are competitors of the cat-people. They knew they couldn't compete with the cat-people on cuteness, so they turned to business and acquisition. They acquired mewonic technology by bribing a cat-technician with a lifetime supply of tuna.

For more evidence of the plot, consider how the dominant candidate for a Grand Unification Theory is String Theory. It was actually first developed by a cat-scientist conducting experiments on a ball of yarn. Whenever you see a cat "playing" with a string, you are actually witnessing one of their routine exercises in science education.

Also consider how every line of discussion in a blog or forum such as CR4 is called a "thread." Another sign of the cat-people.

I have seen a copy of their plan, but only briefly. It is called The Catmunist Meownifesto, written by one of their philosophers, Karl Manx.

Open your eyes, and you will see the lynx to their plan everywhere.

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#24
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Re: Muon Decay: CR4 Challenge (05/12/08)

05/15/2008 6:40 PM

Hi, 3Doug!

Well, you could be correct, except for those authorities I spoke of, and the requirement of creating muons with a separate source of power as compared to mewonic power, which is a source of power, and the immutable high regard Ferengis have for their rules. There is also, come to think of it, those linguistic references, but they might have been based upon a misconception, if indeed there is any misconception being made.

I'm not going to try to differ with you, however. I think "Cats are not seeking to take over in the sense of attacking and conquering as humans would. They seek to make us willingly surrender to their furry cuteness and cuddliness..." is a pretty transparent piece of prose.

I prefer cats as a pet because of their independent low maintenance lifestyles and uncontrollable, constant hunting drive (which requires --in order to remain sharp-- the possession of a powerful ego that transmutes itself into a self-delusional need to believe they are running things). I personally have always regarded their ploys as obvious, and am not, in their vernacular, a 'well-trained' human. Hence, I have been able to retain my objectivity as regards their "cuteness" (not!) and "cuddliness"(prefer my own species, thanks). When I am in the mood, I like them, and I do not for an instant doubt that they are very clever. I think they would like to believe that no humans are like me; but unfortunately (witness the plague), as a species, we are not very outgoing about cats. In fact, we are divided into two distinct camps: those who 'like' cats (sub-dividable into your type and my type), and those who dislike them, for whatever reason. The only time they ever got the edge they wanted was in ancient (by our reckoning) Egypt, and then only by exposing Osiris. (BTW, 'μ' as a chosen symbol for many things in maths & physics originates in Egypt as the word for 'water', not 'cats'. The 'miaow' we attribut to cats is an English-language derivation. Its sound and spelling have widely interpreted variations base on language. It's time they realized that Egypt is in the past.

As to string theory, Wickipedia indicates by: "In 1969 Yoichiro Nambu, Holger Bech Nielsen and Leonard Susskind recognized that [field theories with spontaneous symmetry breaking] could be given a description in space and time in terms of strings. The scattering amplitudes were derived systematically from the action principle by Peter Goddard, Jeffrey Goldstone, Claudio Rebbi and Charles Thorn, giving a space-time picture to the vertex operators introduced by Veneziano and Fubini and a geometrical interpretation ["strings"] to the Virasoro conditions." that, since Albert Einstein, a lot of humans have been doing a pretty good job catching up to the rest of the universe in theoretical physics while maintaining our religion-based paranoic isolationism from it. When our skules begin to marry the theoretical and the practical engineering branches instead of giving the former's hand-me-downs to the latter, we will indeed begin to move forward into the future, catching up to the inhabitants of the other worlds that for now we can only admire from afar.

You and I may agree, however, that cats have encouraged routine and even exceptional educational advances (witness "The Sabre-Tooth Curriculum", by J.A. Peddiwell 1939, the gist of which has been revisited by Hall/Dennis in 'Living and Learning: The Report of the Provincial Committee on Aims and Objectives of Education in the Schools of Ontario', 1968; Ivan Illich's 'De-schooling Society', 1971; and so-called 'free' schools such as the Sudbury School system); a bonafide example of cat contribution to education that comes indisputably from the need for all cats to live free. Even the children's game 'cats cradle' can be instructive --as well as lucrative, as Kurt Vonnegut can attest, although in his 1962 novel there was "no damn cat, no damn cradle" in the string construct...a revelation approaching H.C. Andersen's 'emporer's new clothes' for the 20th century.

But as for the debate about muons being in actuality mewons, I find it difficult to put the two into the same context. I'm afraid I'll have to remain 'silent as a muon' on the subject.

Mark

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#31
In reply to #22

Re: Muon Decay: CR4 Challenge (05/12/08)

05/17/2008 10:02 PM

Hello yet again MarkTheHandyman,

I read the 'moo-onic' bit and was about to put down some 'funny' stuff about Bovine doings, when a few lines further on you had already written it! Do you have to get it right all the time?.................

I like what you have written, but, not being an Engineer, though I have Engineric (is that a word?) (Engineer---ing-ish) (I am always making new ones up. Trouble is, it is me only that knows what they mean. Not good when you are trying to explain stuff right?) leanings , I missed the Math jokes, though, being an 'almost' engineer, I will now search for what you mentioned and I don't know. Thanks. Think its called 'lernin'?

jfmfit

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#25
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Re: Muon Decay: CR4 Challenge (05/12/08)

05/17/2008 12:28 AM

Hello MarkTheHandyman:,

I have a joke which is funny for 2.2 seconds..................same as a muon......

You say it was called a muon after the name of a mother in law, is it because this mother in law was a bit of a silly cow (no offence meant in any way, just a muon joke!)?

jfmfit

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#27
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Re: Muon Decay: CR4 Challenge (05/12/08)

05/17/2008 6:34 AM

Hi, jfmfit!

Pretty cute. However, like 3Doug, you may have confused muons

with moo-ons.

Check out my blog #22 above.

Mark

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#29
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Re: Muon Decay: CR4 Challenge (05/12/08)

05/17/2008 7:21 PM

Hello MarkTheHandyman:,

oh bum!......................I knew I had done something wrong...............actually, I was wondering if the 'joke' was just too obscure but, you at least saw it.

I will look at your blog thanks! And, I secretly knew the peeps who view CR4 are mostly clever enough to 'get' it...................either that or tell me how bad it was!!! LOL!

jfmfit

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#34
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Re: Muon Decay: CR4 Challenge (05/12/08)

05/18/2008 6:27 PM

Hey jfmfit

Please be assured that it was bad and it was obscure enough to be interesting.

As they say in the trade don't give up your day job just yet.

BAB

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#35
In reply to #34

Re: Muon Decay: CR4 Challenge (05/12/08)

05/18/2008 7:52 PM

Hello BlueAussieBoy,

tar ever so! You are a kind lot really you know!

I just look back at some of my attempts of explanation and cringe!!

jfmfit

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#33
In reply to #15

Re: Muon Decay: CR4 Challenge (05/12/08)

05/17/2008 11:15 PM

Hello Mark The Handyman,

I just want to post an apology for my bad explanation in a couple of posts. Missing words mainly.

jfmfit

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#16

Re: Muon Decay: CR4 Challenge (05/12/08)

05/13/2008 7:51 AM

That's pretty straight forward. At relativistic speed, the Muons "life time" is much slower than "our" elapsed time. Ergo, they reach the Earth's surface.

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#17

Re: Muon Decay: CR4 Challenge (05/12/08)

05/13/2008 3:14 PM

http://en.wikipedia.org/wiki/Muons

"Although their lifetime without relativistic effects would allow a half-survival distance of only about 0.66 km at most, the time dilation effect of special relativity allows cosmic ray secondary muons to survive the flight to the earth's surface."

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#18

Re: Muon Decay: CR4 Challenge (05/12/08)

05/13/2008 8:11 PM

It takes 8 GeV of energy in a proton that typically collides with another proton that yields a negative pion. The pion very quickly breaks down into muon. The muon which is 206 times the mass of the electron has the same charge in 2.2 ms it breaks down into an electron and neutrinos. What energy is reatined by the electron and where does the energy go? The las of conservation of mass energy is always though to be operating in that reaction. The electron has much less mass energy than a muon. Neutrino have negligible mass or no mass do they speed away at very high eenrgy?

What is the total equation?

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#32
In reply to #18

Re: Muon Decay: CR4 Challenge (05/12/08)

05/17/2008 10:21 PM

Hello pron7,

I know this may sound stupid but, how is it possible to know the energy made/or taken when a muon hits a pion?............Or something like that.

Do you work with thousands or millions of them and work back until you have the answer as 1 Billoin Electron Volts? It would not takes many to fuel power for the whole human race!

jfmfit

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#19

Re: Muon Decay: CR4 Challenge (05/12/08)

05/14/2008 4:52 PM

Don't colliding particles propogate thru the atmosphere?

and doesn't this propogation (looking like a inverted family tree) eventually get within 650 meters of the earth?

Therefore, doesn't the initial molecular excitation transvere via some good vibrations to our fishing buddies on terra firma?

?

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#37
In reply to #19

Re: Muon Decay: CR4 Challenge (05/12/08)

05/21/2008 1:56 PM

Hi, Guest!

In the case of ionic excitation due to a large electrical interference burst, the valence of some molecular structures is disturbed when oxygen ring electrons (the weakest ring in many valences) may be sent flying into other oxygen ring electrons to knock them out of orbit, thus forming a chain reaction called an ionization avalanche.

Due to the hunger of oxygen to unite with whatever is nearby, this disturbance resettles itself quickly with changes in nearby molecular structure reflecting fewer or more oxygen parts as they neutralize their temporary charges.

The initial charge burst has to be very large (eg storm clouds, lightening bolts, Van DenGraf generators, etc.) to effect such an avalanche. Muons couldn't cut it, I don't think, even with their heavily raining down on earth.

Mark

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#38
In reply to #37

Re: Muon Decay: CR4 Challenge (05/12/08)

05/22/2008 1:56 PM

Hello MarkTheHandman,

just like to say that's the first time I have seen 'ionization avalanche' explained. Though I can't really remember the powers now, I was helping my Niece when she was asking for info on how valances work and what kind of power causes the splitting of the atom's valance, this was about 18 years ago, but the power that holds just about all atoms together is phenomenal! And, though I found it hard to 'grasp' when I first read it, is of course included atoms in solids as well as gaseous and liquid situations. I still find it difficult to imagine how something so tiny can have such a huge force?

jfmfit

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#20

Re: Muon Decay: CR4 Challenge (05/12/08)

05/14/2008 10:24 PM

Muons, Clingons, Hobbits, it's really all speculation

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#36

Re: Muon Decay: CR4 Challenge (05/12/08)

05/21/2008 1:03 PM

Irritating, to say the least.

I am willing to tolerate all those posters who point out that John said (or should have said) "half-life" instead of life-time, but all the time-dilation stuff is unacceptable. "Oh, I meant 2.2/((c-v)/c) instead of 2.2" went out when "Women's Lib" kicked out the "stupid woman" jokes.

Please can the moderators exercise some discipline on "too clever-by-half" problem setters?

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Anonymous Poster
#39

Re: Muon Decay: CR4 Challenge (05/12/08)

07/10/2008 9:18 AM

To elaborate on the correct answer above, imagine you travel with one of the muons after it had been created. If it it is moving towards the Earth at a speed close to the speed of light, the distance to the Earth's surface in this frame would be much less than 100km due to Lorentz-Fitzgerald contraction. You and the muon are able to hit the Earth's surface in a time which is small enough for the muon to have not yet decayed.

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