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mikelewis Donating Member (1000+ posts) Send PM | Profile | Ignore Wed Mar-09-05 11:19 PM
Original message
I have a Math question about the Pentagon
This level of math is well above my understanding so I'm not going to even attempt it however, I think this one math problem could put the issue of what hit the Pentagon to rest.

I'm sure, with as much attention that we place on missiles, we have a formula that can determine an estimated projectile impact penetration into a hardened structure. Since flight 77 weighed X tons and was traveling at y miles an hour, what is the maximum penertration depth this projectile could obtain into Z feet of a hardened concrete structure?

If it possible that the plane could have penetrated that deeply, though still highly skeptical, I'll be satified that it was flight 77 that hit the pentagon. Is this fair?

If it turns out that it was impossible for the plane to penetrate this deeply into a hardened concrete structure, what other alternatives exist that could explain what happened?

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SkyIsGrey Donating Member (288 posts) Send PM | Profile | Ignore Wed Mar-09-05 11:41 PM
Response to Original message
1. You would also have to
take into account that the plane is not a solid object. Being hollow, and able to collapse under impact stress, thus absorbing an X amount of impact energy, would have to be included into the calculation.
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mikelewis Donating Member (1000+ posts) Send PM | Profile | Ignore Thu Mar-10-05 12:12 AM
Response to Reply #1
2. Let's just for arguement sake....
Edited on Thu Mar-10-05 12:13 AM by mikelewis
Say that it collapsed into a crumpled ball of aluminum with a diameter of 14 feet before it struck the wall. Also, crumple the weight of the wings into the ball. I believe the wings should have sheared off but some believe they folded into the fuselage. To me this sounds absurd but hey, some people believe a plane made that hole. I'm just asking them to prove it with math.

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DulceDecorum Donating Member (1000+ posts) Send PM | Profile | Ignore Thu Mar-10-05 01:56 PM
Response to Reply #2
3. No takers
because then they might be called upon to explain how the very same set of circumstances crashed BOTH towers.

Whenever they say that the plane is hollow, they are talking about the Pentagon. I almost never hear that argument presented in WTC discussions.
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mikelewis Donating Member (1000+ posts) Send PM | Profile | Ignore Thu Mar-10-05 04:36 PM
Response to Reply #3
6. Give them some time... This is not an easy problem
Truthfully, I want to believe it was Flight 77 that hit the pentagon. I want to be completely mistaken about everything I believe about that day. If they can convince me using a mathematics then I will leave the issue alone. I personally don't think a plane could have created this hole, let alone penetrate as deeply as it did but the latter can be solved as a function of X.
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spooked911 Donating Member (1000+ posts) Send PM | Profile | Ignore Thu Mar-10-05 02:39 PM
Response to Original message
4. I think the problem is that we don't really know the strength of the
Pentagon wall. So even if we can calculate the potential force of flight 77, we don't know how such a force would affect the wall, which was supposed to be hardened and reinforced.

My main problem with the pentagon hit is this:

either the wall strongly resisted the impact of the plane in which case the plane should have blown up outside (as shown in the security camera video) and thus left a lot more debris outside

OR

the wall gave way and let the plane in, which doesn't explain the huge explosion seen in the security camera video and it doesn't explain the minimal size of the entry hole.

I don't see how you can have a huge explosion outside as the plane hits AND also have the plane essentially disappear completely into the building.

I also don't see how a 757 could go into the building without knocking down those huge cable spools that were just outside the impact area.
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mikelewis Donating Member (1000+ posts) Send PM | Profile | Ignore Thu Mar-10-05 04:31 PM
Response to Reply #4
5. We know the composition of the wall and I'm sure there are...
studies all ready available that can guage a projectile impact penetration upon a reinforced concrete structure.

How deep did the object penetrate into the structure?

What is the thickness of the walls over the distance the penetration?

What was the estimated air speed at impact?

What was the maximum weight of the aircraft with a full fuel tank, passengers, luggage etc...?

Perhaps the walls of the pentagon are engineered to absorb a projectile hit and this is why the damage wasn't as severe as one would assume. Maybe it's designed in such a way as to minimize the effect that a projectile would have over a large area by collapsing in upon itself and distributing the kenetic energy through some sort of internal shock absorber. This wouldn't explain what happened to the wings but it could explain the size and depth of the hole.

Forget for a second we are trying to recreate a plane strike on the pentagon and form the mass of the plane into a crumpled ball of aluminum. Forget it's even the pentagon and just use the material, the weakest material. How far could this object penetrate into wood, or concrete or steel? There must be a formula in existence that can answer this question.






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spooked911 Donating Member (1000+ posts) Send PM | Profile | Ignore Thu Mar-10-05 05:11 PM
Response to Reply #5
7. Yes, I'm sure there is a formula
but I am not a physicist or engineer so I don't know it.

I'm sure you are familiar with the Sugano experiment where an F4 jet was slammed into a concrete wall.

http://www.911-strike.com/missing-confetti.htm

The plane shattered into mostly small pieces and barely left an indentation on the concrete wall.

From the 9/11 strike site:
"Due to the many complexities of the structural analysis of such a crash, the exactly "correct" impact hole size and shape is simply an imponderable for both the crime scene investigator and the perpetrator alike. What we are saying, is that the hole in the Pentagon does appear (at least in its basic size and shape) as a plausible outcome of a 757 crash, although questions do remain about the specific nature of damage to the columns and the interior features of the Pentagon offices."

And here is an alternative view-- Karl Schwarz's article saying an A-3 hit the Pentagon, firing a missile first.

http://www.onlinejournal.com/Special_Reports/020205Schw...
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mikelewis Donating Member (1000+ posts) Send PM | Profile | Ignore Thu Mar-10-05 08:25 PM
Response to Reply #7
12. Good point about the concrete columns but ....
They can be deleted from this equation to make it easier and just use the outer walls of the rings.
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AZCat Donating Member (1000+ posts) Send PM | Profile | Ignore Thu Mar-10-05 06:39 PM
Response to Original message
8. It is, unfortunately, far more complicated than that
This is difficult to determine, unfortunately, and cannot be modelled by a simple equation as you have theorized. There are software programs that can model and simulate such an impact, but even they have limitations.

The difficulty arises from the nature of the problem - it is a high-speed impact between two non-homogeneous deformable objects. There are assumptions one can make to get a rough idea of the force required to "punch" through a wall, but the uncertainties of such an estimate would be large IMHO.
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mikelewis Donating Member (1000+ posts) Send PM | Profile | Ignore Thu Mar-10-05 07:23 PM
Response to Reply #8
9. If they can model canon shot penetratrion the same equation should apply..
Here's a link to several different type of impact equations based upon different types of steel. I'm sure, the same thing exists for impact probabilities into other types of material.

http://www.combinedfleet.com/formula.htm
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AZCat Donating Member (1000+ posts) Send PM | Profile | Ignore Thu Mar-10-05 07:40 PM
Response to Reply #9
10. No.
Edited on Thu Mar-10-05 07:41 PM by AZCat
These are two different cases - the example you point to does attempt to establish some sort of a probabilistic relationship between penetration depth and other variables, but it is based on several assumptions that do not translate well to the case of the Pentagon impact.

The Pentagon impact defies simplification because of what I said above: the two objects involved are both non-homogeneous and deformable. In the shot penetration case, the impacted surface is composed of one single material (i.e. it is homogeneous) and the impacting object is also a single material, along with being a simple shape. Impact dynamics of thin-walled shells such as aircraft are notoriously complicated and cannot be simplified easily (if at all). There also seems to be a great deal of empirical observation used to calibrate the equations in the shot penetration case, whereas there is little data available for comparison in the Pentagon case.

If you want to make some estimates, go ahead - but they won't be very reliable.



Edit: spelling
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mikelewis Donating Member (1000+ posts) Send PM | Profile | Ignore Thu Mar-10-05 08:21 PM
Response to Reply #10
11. I understand this but I am simplifying it and stacking the deck...
in favor of those who believe that 77 hit the pentagon.


I'm sure, somewhere, someone knows how to determine if a X ton projectile can in fact penetrate Y feet into a hardened concrete structure. I'm sure a formula exists that can predict the outcome within a certain tolerance. I have come across many that deal with the penetration into armored steel plating and also into human tissue. I would be willing to bet money that there have been studies performed on hardened structures. I am positive that physics applies to this scenario even if there is a error threshold. The shot penetration case is simply one example I've come across. There are others. The math is beyond me but I figure that there are some pretty smart people here, who have very strong opinions. I would think this would have been the first question asked.

Is it possible for the plane to have penetrated as deeply as it did? Forget about the lack of video evidence and the lack of adequate photographic evidence. This can be dabated till the cows come home. The physics behind the event will tell us more than any picture.

As I stated, this level of math is beyond me but I'm sure some of the more educated on this board will most certainly be able to prove that it is physically possible for this plane to make this hole.
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AZCat Donating Member (1000+ posts) Send PM | Profile | Ignore Thu Mar-10-05 08:47 PM
Response to Reply #11
13. Of course physics applies to this scenario...
Why wouldn't it? :shrug:

This is an area with which I am not unfamiliar. I do not think you understand the differences between the two cases: a X ton projectile impacting a homogeneous concrete wall and a thin-walled shell (like an aircraft) impacting a non-homogeneous wall. This question has been asked a number of times (both here and in the engineering community) and there have been a number of tests to verify the algorithms that are used to model such impacts. It is certain that complex impacts cannot be modelled by an analytical equation such as you are proposing, but must be modelled using numerical methods. If you are interested in learning about those methods (and the algorithms used to model complex impacts) there is an amazing wealth of information on the internet, but unfortunately there are few freeware programs available. I posted some links on another thread where Old and In The Way asked a similar question but most of the info/tutorials are for the popular programs such as ProEngineer, MS-DyTran, ANSYS, etc.
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gbwarming Donating Member (1000+ posts) Send PM | Profile | Ignore Thu Mar-10-05 08:58 PM
Response to Reply #11
15. The constants might be developed given a supply of pentagons and 757's
Edited on Thu Mar-10-05 09:14 PM by gbwarming
Naval ordnance folks need to know if their shells and armor will work so they do thousands of tests to determine these constants that go into the various formulae. It's a major effort to do that kind of work and I'm not aware of any similar study for penetration of aircraft sized aluminum balls into vaguely defined hardened concrete structures.
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mikelewis Donating Member (1000+ posts) Send PM | Profile | Ignore Thu Mar-10-05 08:51 PM
Response to Original message
14. Here's what 911physics.org has to say
http://physics911.ca/modules/news/article.php?storyid=1...

"Lets do some comparisons with weapons specifically made to penetrate strong buildings.

During WW 2, the British developed the Tall Boy Bomb

* http://www.members.aol.com/nukeinfo2 /

It weighed 12,000 lb and could punch its way through 10 ft of steel reinforced concrete, when dropped from a great height ( a Lancaster bomber) Very impressive! The Pentagon may not be as strong, but the engine is alleged to have punched through nearly triple this width. The engine weighs about the same as the "Tall Boy." However the "Tall Boy" was travelling at several times the speed, and also contained explosives. And yet, somehow, it appears to be only marginally more effective, perhaps even less. All that engineering for nothing! If the penetrative performance of the 757 engine is anything to go on, it seems that the Brits would been better off to save their money and just drop big lumps of scrap metal. Not learning this lesson, they went on to develop the heaviest bomb of WW 2 , the 22,000 lb Grand slam bomb which could penetrate steel reinforced concrete to a depth of about 12 ft. In addition to its enormous weight and explosive power, it was dropped from Lancaster bombers, giving it great speed by the time of impact. The article mentions that the bombs were exceeding the speed of sound (760 mph), by the time they hit, but doesnt mention by how much. That could be calculated if you knew the height at which the Bombers were flying.

* http://www.accessweb.com/users/mconstab/bombs.htm (also see previous link)
"

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AZCat Donating Member (1000+ posts) Send PM | Profile | Ignore Thu Mar-10-05 09:04 PM
Response to Reply #14
16. The analysis at the first link loses me...
when it makes this transition:
It appears to be at angle of something like 45 degrees from the first ring damage area. How much stone has been penetrated to make this hole? Assuming no internal walls, 6 walls. If each wall is 3 feet thick, thats 18 ft of stone, plus or minus any inaccuracy of the guess of the thickness of the walls, perhaps plus anything that was in the way, inside the buildings. Because its at a 45 degree angle, whatever has made it has actually had to punch out 27 ft of stone.
</snip>

There is significant difference between six separate walls three feet thick and one wall eighteen feet thick. While I applaud those trying to understand the admittedly confusing events of September 11th, I wish there were less "certainty" in their statements when they are a bit casual with their mechanics.
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gbwarming Donating Member (1000+ posts) Send PM | Profile | Ignore Thu Mar-10-05 09:22 PM
Response to Reply #16
17. Worse than that - there are no concrete walls between entry and exit
Why in the world did they choose to assume six, three foot thick walls? The rings are connected on the first floor. Right?
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AZCat Donating Member (1000+ posts) Send PM | Profile | Ignore Thu Mar-10-05 09:34 PM
Response to Reply #17
18. I don't know.
I've just gone back and looked at the ASCE Pentagon report, and I don't see where the six walls come from either. Perhaps someone would kindly enlighten me to the source.
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mikelewis Donating Member (1000+ posts) Send PM | Profile | Ignore Thu Mar-10-05 09:49 PM
Response to Reply #18
19. Three rings pierced - six outer walls of 16" thick
Edited on Thu Mar-10-05 10:06 PM by mikelewis
http://911research.wtc7.net/mirrors/guardian/Pentagon/w...

"It is also claimed that the Pentagon walls were specially reinforced. However, the outer Pentagon wall comprised a framework (grid) of 10 inch reinforced concrete members with the intervening space filed with 8 inch thick brickwork. Over this was placed about 6 inches of decorative limestone. So, the (outer) wall was at most, 16 inches thick, and was not particularly strong (contrary to media reports). The following photos show this to be the case."






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AZCat Donating Member (1000+ posts) Send PM | Profile | Ignore Thu Mar-10-05 10:00 PM
Response to Reply #19
20. Uh, that seems problematic.
1) Six times eighteen inches is 108", or nine feet, not eighteen. And this cannot be simplified as one wall of the same thickness, as I stated in post #16.

2) Where do you get the eigteen inch thick walls from? I have looked through the ASCE report and have not found confirmation of that.
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mikelewis Donating Member (1000+ posts) Send PM | Profile | Ignore Thu Mar-10-05 10:07 PM
Response to Reply #20
21. Sorry, I was going from memory at first then thought I should back it up
It was 16" not 18"
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AZCat Donating Member (1000+ posts) Send PM | Profile | Ignore Thu Mar-10-05 10:13 PM
Response to Reply #21
23. That's okay.
Thanks for correcting/documenting.
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LARED Donating Member (1000+ posts) Send PM | Profile | Ignore Thu Mar-10-05 10:10 PM
Response to Reply #20
22. All the walls
were not reinforced concrete. As you can see it appears the inner wall is a simple brick construction that is far less than 18" thick. I think the 9 feet of wall and the wall being all reinforced concrete are one of the many internet "facts" abounding in the 9/11 ever widening mythology


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AZCat Donating Member (1000+ posts) Send PM | Profile | Ignore Thu Mar-10-05 10:19 PM
Response to Reply #22
24. Some of this stuff is hard to explain, though.
Without some sort of common base of knowledge it is difficult to get across certain points, like the one I have been making here - that six walls of a certain thickness are not equivalent to one wall (with a thickness equal to the sum of the six walls) when considering impacts.
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mikelewis Donating Member (1000+ posts) Send PM | Profile | Ignore Thu Mar-10-05 10:38 PM
Response to Reply #24
27. Yes AZ, I understand that point and I'm pretty sure there is some way to
solve this riddle.

If I drop a 10 pound ball into sand from the same height, wouldn't the ball sink into the sand the same distance assuming you hit sand with the same overall consitancy.

Let's say that that ball crashes through 16" of sand and then there is a 20 foot empty space. Let's say we launched the ball so hard it penetrated 6 of these sand walls. The amount of force necesary to do this is an equation.

The sand wall cannot absorb as much kenetic energy as sand that was 9 foot thick but ultimately, the ball will have to travel through a total of 9 foot of sand to reach it's final resting place. I'm sure there is a formula that can tell you what sort of ordinance you need to penetrate a building and reach a particular floor. This formula would take many of these variable into account.

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LARED Donating Member (1000+ posts) Send PM | Profile | Ignore Thu Mar-10-05 10:45 PM
Response to Reply #27
29. In theory you are correct
Edited on Thu Mar-10-05 11:12 PM by LARED
The problem is that most of this type if work depends on empirically found relationships that get turned into mathematical formulas. The engineering world is full of them.

Why?

Taking your example of sand. Someone somewhere can easily pound balls into sand at different velocities and energies to determine a relationship between sand as a media and the energy of the ball impacting it. Using this data they develop empirically derived equations and rules of thumb as too how deep it will travel given a energy level. To determine an equation based on the ball acting on the sand is quite difficult. Newtonian mechanics are great when dealing with solid objects that can be easily idealized. When you start dealing with something like a ball hitting sand or a plane hitting a single wall the variables start piling up making the derivation of an equation quite difficult if not impossible.

No one is going to crash jets into buildings to determine the relationship between the energy of the jet and the amount of of jet that is destroyed.
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AZCat Donating Member (1000+ posts) Send PM | Profile | Ignore Thu Mar-10-05 11:09 PM
Response to Reply #27
31. I didn't mean to imply...
that you were having difficulty with the point. I apologize if it seemed like I was. There have been posters that have had difficulty with this sort of stuff and I should have been clear that you haven't. It's frustrating when it happens so I was just taking the opportunity to grouse about it - sorry.

The sand analogy fails because sand is a particulate and cannot support shear stresses, while a wall of material such as metal or concrete can translate the stresses from one area to another. The width of a particular wall is important when determining how it reacts to a impacting object, so there is no equivalency between three walls of one foot thickness each and six walls of six inch thickness each.

There is no formula because as LARED (in post #29) and gbwarming said (in post #15) there is little data available to produce such empirical relationships. It is possible to do it with numerical simulations - the auto industry discovered this, used their data from crash tests to validate their models and "virtually" crashed cars hundreds of thousands of times - but one needs to have collected empirical measurements first in order to have a successful simulation. NASA/NIST/everybody else has been working to do this - the CID, the crash mentioned above (F-4 into concrete wall), some hundred drop tests have all been done do help validate the algorithms used in the software packages I mentioned above.
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mikelewis Donating Member (1000+ posts) Send PM | Profile | Ignore Thu Mar-10-05 10:23 PM
Response to Reply #22
25. It appears that the columns were steel re-inforce concrete - not the walls
http://concreteproducts.com/mag/concrete_nothing_secret... /

"They noted that the structural performance of the Pentagon a five-story fortress whose 1941-1943 construction saw placement of more than 400,000 yd. of concrete validates measures to reduce collapse resulting from unlikely events, including 1) continuity, such as the extension of floor reinforcements through structural supports; 2) redundancy in design and construction, such as two-directional framing of the floors; and, 3) spirally steel-reinforced concrete columns that absorb energy generated from a lateral load, such as high wind, earthquake or aircraft impact."

Okay, get rid of the "steel-reinforced" anything and just deal with brick and mortar. How many walls of 16" thick brick and mortar can a plane weighing x tons penetrate traveling at y miles an hour?
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LARED Donating Member (1000+ posts) Send PM | Profile | Ignore Thu Mar-10-05 10:37 PM
Response to Original message
26. A usable equation is fairly simple
Edited on Thu Mar-10-05 10:48 PM by LARED
But the math is not even feasible as far as I know.

The equation is;

The kinetic energy of flt 77 {(1/2MV^2) = Change in kinetic energy (Work) of the building + Change in kinetic energy of the plane (Work)}

Work is the mechanical energy produced by or applied to the process or change in kinetic energy

The kinetic energy of the plane is known. The change in energy or work done to the plane as it was destroyed is not known. The change in energy or work done to the Pentagon as it was destroyed is not known.

As far as I know there is no mathematical equation that is going to provide you with an answer. The work processes at play are far to complex to be expressed in an equation.

If I recall correctly, the pentagon report indicates they have developed some numerical model to study the effects of the impact on a column to determine if the "fluid mass" of the jet could destroy a column. As I recall it indicated that this "simple" numerical model matched well with the observed condition of the columns.
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mikelewis Donating Member (1000+ posts) Send PM | Profile | Ignore Thu Mar-10-05 10:43 PM
Response to Reply #26
28. What would be the total kenetic energy of the plane at the time of impact?
Edited on Thu Mar-10-05 10:44 PM by mikelewis
Wouldn't this be a function of mass and velocity?
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LARED Donating Member (1000+ posts) Send PM | Profile | Ignore Thu Mar-10-05 10:47 PM
Response to Reply #28
30. Yes
it's 1/2 X mass X velocity^2
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mikelewis Donating Member (1000+ posts) Send PM | Profile | Ignore Thu Mar-10-05 11:39 PM
Response to Reply #30
34. I need help with the math.
A fully laden Boeing 757-200 has a mass of 115,895kg {1} and a max speed of 914km/h {1}.

What would the Kenetic energy be without consideration for the explosiion of Jet Fuel?

Now, what would be your best guess as to the impact the explosion of jet fuel had on the kenetic energy in the lateral movement of the plane through the building. I would think it would be minimal but what do I know. Let's just add 10% additional kenetic energy to the movement of the plane as it travels through the building.

How far did the plane travel through the building?







{1} http://www.airliners.net/info/stats.main?id=101

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AZCat Donating Member (1000+ posts) Send PM | Profile | Ignore Thu Mar-10-05 11:54 PM
Response to Reply #34
36. I have done similar calculations before...
In a previous post, I calculated the equivalent mass of TNT for a Boeing 767 travelling at 400 kts. The numbers may differ, but the math is the same. I will do it with your numbers.

Boeing 757-200 calculation:

m = 115,895 kg
v = 914 km/h = 254 m/s

KE = 0.5*m*(v^2) = 0.5*(115,895)*(254*254) = 3.73 GJ (GigaJoules)

This is the equivalent of approximately 1,780 lbs of TNT.


As to the fuel, I would have to look up the specific energy of combustion of kerosene (I had it written down but I don't know where).
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mikelewis Donating Member (1000+ posts) Send PM | Profile | Ignore Fri Mar-11-05 12:14 AM
Response to Reply #36
37. So the total resistance through the area travelled...
Edited on Fri Mar-11-05 12:17 AM by mikelewis
has to equal this number plus whatever energy the fuel combustion added?


Now according to this image provided by the Washington Post


The plane traveled through six walls consisting of 10" of reinforced concrete with a brick center and a 6" of limestone facing and through 33 steel reinforced concrete pillars. Each of these things has a certain value of resistance that should equate to roughly 3.73+ GJ of negative Kenetic Energy. Does this make sense?

http://www.washingtonpost.com/wp-srv/nation/graphics/at...
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AZCat Donating Member (1000+ posts) Send PM | Profile | Ignore Fri Mar-11-05 12:27 AM
Response to Reply #37
38. I thought it hit the second-floor slab...
along with some objects outside.

But yeah, you have the right idea (I think). The problem lies in assigning how much energy was used on each column. The same force applied at different places on a column will have different results, so a column that displays more damage will not necessarily have used more of the "energy budget" of the aircraft.

This is where numerical simulations, like the ones the NIST is using for the WTC, are useful. They can simulate the collision (several times, if necessary - like in Monte Carlo methods) and can give an idea of how the energy dissipated throughout the structure.
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mikelewis Donating Member (1000+ posts) Send PM | Profile | Ignore Fri Mar-11-05 01:21 AM
Response to Reply #38
39. See my thinking is....
This was a massive amount of energy to be directed against the Pentagon. While it is possible that it was flight 77 as you clearly believe, I am not quite there, yet. If the impact had a kenetic enegry equal to 1700 sticks of TNT, I would think that the impact area would have been more conical in shape. From the photots I have seen, it appears that the plane simply burned a hole right through the building.

From this picture, it seems as though the plane did strike the second floor slab.



On this picture we see that the plane had sufficient kenetic energy to punch clear through to the outer wall of the C ring. I have not been able to find the actual distance traveled but so far I'm estimating to be about 220 feet.



But for our calculations, let's just assume it went straight through without stiking the slab. This will skew the result in favor of the OCT anyway so let's just leave that out.

I understant that the strength of concrete is measured in PSI. How does this translate in to keneitic resistance? Let's assume for a second that the concrete had roughly 4000 psi of resistance. What would the amount of kenetic resistance be in this type of material?


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AZCat Donating Member (1000+ posts) Send PM | Profile | Ignore Fri Mar-11-05 01:42 AM
Response to Reply #39
40. I'm too tired to tackle this tonight.
I have spring break next week. Even though I am working a few days, I will try to post some sort of "primer" for calculating this sort of thing.

I am warning you though - it probably won't give you the answers that you want. This stuff is pretty sensitive to conditions (where the force is applied to a column, etc) but if you want to know how this stuff is calculated I will try to write something up.
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AZCat Donating Member (1000+ posts) Send PM | Profile | Ignore Tue Mar-15-05 09:28 PM
Response to Reply #40
50. I'm sorry - I'm not going to do this this week.
I thought I'd have more free time this week but I have had to work on a couple of projects, and I forgot that the NCAA tourney starts Thursday. I'm not going to be able to do anything for this until next week at the earliest.

A recommendation I would make regardless is that you go to a used bookstore and pick up a either a book on statics or mechanics of materials (or both). These texts go over the fundamentals of what forces do to materials and how to model them. There's much more, but those two topics should give you a good introduction.
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gbwarming Donating Member (1000+ posts) Send PM | Profile | Ignore Fri Mar-11-05 03:01 AM
Response to Reply #39
43. You need to know how far the structure deflects as the load is applied
Mechanical energy is measured in Foot-Pounds or Newton-Meters (also called Joules): Force times distance. So to find the energy required to damage the building (and the plane!) you need to find out the history of force over the distance.

The material strength is only part of the information. The graph below shows a typical stress vs. strain diagram for a ductile material like steel or aluminum set up to be tested in a tensile (pull) test - as you pull on it it initially strains (stretches) in proportion to the load (this is the straight line on the left) and then as the load is increased a little more the material continues to stretch until it eventually breaks. The area under this curve is proportional to the energy required to break the material (by pulling). The columns aren't as simple to analyze because rather than a direct pull they are loaded by bending, compressive and shearing forces AND they're a composite of steel and concrete.

Most engineering design is only concerned with the straight line portion of the curve because up to that point the material will spring back to its original shape when the load is removed (elastic deformation) but beyond that the material is permanently deformed (plastic deformation). The elastic part is also easier to deal with because it's linear. Brittle materials like concrete just snap when they get to some maximum load (different in compression and tension). Reinforced concrete behaves in a much more complex way than either material alone. There are conventional techniques for analyzing reinforced concrete structures, but I don't know of anything simple that deals with behavior beyond the point where the concrete begins to fail.

Sorry, I know that's not an answer to your question. If these were very simple shapes of one type of material, and we were only going in the elastic range this would be an easy problem.

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Christophera Donating Member (1000+ posts) Send PM | Profile | Ignore Thu Mar-10-05 11:11 PM
Response to Original message
32. Before Math; Examine all factors, Seek Order
Edited on Thu Mar-10-05 11:13 PM by Christophera
Of course its a challenge to see if such a collision consistent with damage done can occur as defined by physics.

I think we know the calculations can't be produced. To produce an understanding I advocate a ordered summary to evaluate cause/effect in the collision process.

A plane; mostly air with fuel and cargo, having only 2 large hard projectile elements, engines; hits at say 400 plus in level flight but somwhat oblique, 27 degrees off perpindicular.

A building, recently hardfaced with anti ballistic shielding. Unreinforced concrete panel walls and masonry facing are underneath. Heavy steel reinforced columns holding up beams of the same stand in the subfloor areas.

Collision; The planes nose and right wing tip connect tearing off pieces the plane enters just below the first floor. The soft nose mushroms to the side and slides a tiny bit down the face of the building. All components in motion share the same vector into vertical walls and floors. Soft aluminum forms around the floor but break big chunks off plowing them in front or throwing them off. Fuel is vaporized and catches fire. The heavy titanium and hardened steel engines with components turning at high RPM drag engine mounts and shreds of wings into the building. The panels of concrete hit with soft forming metals collapse, fall. The soft metals form and roll into, over and through the collapsing elements of structure.
The engines begin tumbling out front of the slowing fuselage, reacting to the line of travel and impact vector with each fresh impact, spinning and bouncing through sub floor areas.

If they hit a non steel reinforced wall at the end of that kind of travel, only beams columns and floors were steel reinforced, they might penetrate, but would leave irregular holes at their reduced speeds as they deflected more with the impact vector in slowing.

All of the above is contradictory to the damage to the face and the exit hole left in the building which has a very unform ballistic quality, and there is only one hole.

In all we see the soft aluminum plane making a large hole because it does not penetrate, it spread out and knocks things down. Shallow penetration over a wide area, except for the engines.

A supersonic missile with a hardened airframe would do almost exactly what we saw. Others define a small plane launching a missile from the parking lot video. Since a missile with a charge in it blowing up causing the explosion seen in the video would be destroyed before reaching the wall where the exit hole is, the explosion was caused by stationary charges placed near a fuel tank/storage area on the ground level of the pentagon set off with a vibration detector on the face of the wall.

The sound barrier was broken that morning and witnessed by an entire school yard and many residents.

Conclusion; (NOTE: Pictures show guys in suits carrying an obect on a wooden pallet covered with a blue tarp.)

A bunker buster AND hardface anti ballistic building applique was tested in a giant ruse along with a bunch of mkultra pentagon witnesses to add a military target to 9-11 and more justification to make war.

SOON TO BE RELEASED ON VIDEO! (We do hope those citigo and sheraton tapes leak)
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LARED Donating Member (1000+ posts) Send PM | Profile | Ignore Thu Mar-10-05 11:23 PM
Response to Reply #32
33. I love asking this question
How do you know this?

All of the above is contradictory to the damage to the face and the exit hole left in the building which has a very unform ballistic quality,.....

And

In all we see the soft aluminum plane making a large hole because it does not penetrate, it spread out and knocks things down. Shallow penetration over a wide area, except for the engines.

BTW, aluminum is not soft.
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Christophera Donating Member (1000+ posts) Send PM | Profile | Ignore Fri Mar-11-05 02:07 AM
Response to Reply #33
41. Compared To Concrete, Aluminum Is Soft.
I cut an aluminum airstream trailer in half with my pocket kife and crush soda cans with my bare hands. My knife is stainless steel.

Aluminim is light for its strength and certainly there are some harder aluminum alloys for machining, but it sure ain't classed as hard.
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mikelewis Donating Member (1000+ posts) Send PM | Profile | Ignore Thu Mar-10-05 11:49 PM
Response to Reply #32
35. Welcome to DU -
I agree with you on what hit the Pentagon but I disagree with your assessment that the physics cannot be recreated, at least in a general sense.

We know the starting Kenetic energy and the final kenetic energy, 0. If we can find resistance data on the columns and the walls, we can find out if the plane had enought kenetic energy to travel that distance through that resistance. I don't believe it did and that's why I'm posting this. Until the Government exposes a few more details, like the video tapes and photos of the crash site, I will be highly speculative.
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Christophera Donating Member (1000+ posts) Send PM | Profile | Ignore Fri Mar-11-05 02:22 AM
Response to Reply #35
42. Basis For Extrapolation
Edited on Fri Mar-11-05 02:23 AM by Christophera
With the variety of the columns impacted, wieghting each one relating to a perfect impact scenario, could come up with a factor that could float either way to see minimums and maximums, but the impact vector and the deformation of the aluminum combine into a severly difficult equation,

Perhaps the above average concrete density could be matched with a an average plane density having a steel component proportionately sized inside of a hollow aluminum test projectile fired from a 45 caliber pistol with about a 30% powder load reduction into concrete, 600 foot per second. Then the failing of the walls becomes too complex to simulate because your test concrete is a solid block. Many unknowns but with some known factors you might see that you were close to knowing or showing the possibility or not by the math.
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spooked911 Donating Member (1000+ posts) Send PM | Profile | Ignore Fri Mar-11-05 08:21 AM
Response to Reply #32
44. Great synthesis! Questions though:
1) what do you think was being carried on the pallet covered with a tarp???

2) what on earth is "hardface anti ballistic building applique"?

3) are you saying explosives were set where the "exit hole" was? Why would they do that?
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Christophera Donating Member (1000+ posts) Send PM | Profile | Ignore Fri Mar-11-05 04:43 PM
Response to Reply #44
45. Bunker Buster Airframe
Edited on Fri Mar-11-05 04:45 PM by Christophera
Spooked911 wrote:

"1) what do you think was being carried on the pallet covered with a tarp???"

Personally I think it is the hardened airframe of the supersonic bunker buster prototype missile that made the exit hole. We have no information on it other than the approximate size and perhaps weight, pure speculation within a scenario.

Spooked911 wrote:

"2) what on earth is "hardface anti ballistic building applique"?"

Anytime any structure has materials applied to it that are intended to harden or toughen it the term "hardface" is applicable. In this case the intent was to protect from projectiles, or anti ballistic The remodel of the pentagon included such work

Spooked911 wrote:

"3) are you saying explosives were set where the "exit hole" was? Why would they do that?"

No explosive were set at the exit hole. Only at the impact area to create the fuel bomb we saw on video. They wouldn't set charges at the exit hole because it would foul the results of the ballistics test of the missile airframe impacting the exit hole wall and it's penetration/deflection potentials, vital considerations in the design of bunker busters. We all know how obsessed with those certain officials are.

We have a secrete element of government camoflaged within the intelligence community and elsewhere.

Here is an appropriate motto under these conditions,

SAVE THE GOOD PEOPLE IN THE US GOVERNMENT, NOW! (If you want a future in America.)
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spooked911 Donating Member (1000+ posts) Send PM | Profile | Ignore Fri Mar-11-05 11:19 PM
Response to Reply #45
47. Okay-- makes sense. Thanks!
What do you think of Shwartz's theory on the
A3 skywarrior hitting the Pentagon?

btw, I like your motto-- very fitting.
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Christophera Donating Member (1000+ posts) Send PM | Profile | Ignore Sat Mar-12-05 02:13 AM
Response to Reply #47
48. A3 As Missile Carrier?
Perhaps, is it supersonic? Doesn't look it.

I liked his video frame breakdown but the profile is quite a bit different from the A3.
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spooked911 Donating Member (1000+ posts) Send PM | Profile | Ignore Wed Mar-16-05 08:06 AM
Response to Reply #48
52. Rebuttal to the A-3 theory
(Russel Pickering wrote this and sent it to me via e-mail)

COMMENTS ON MYTHOLOGY BUBBLE PART 5

I thrive on all points of view regarding 9/11 and realize that every idea put forward will eventually benefit the discovery of truth. It is not normally my style to critique anybody else’s theory. The reason I have chosen to do so resides in this one statement, “The information in this article has been hand-delivered to New York Attorney General Eliot Spitzer.”

Here is Karl’s article for reference: http://www.onlinejournal.com/Special_Reports/020205Schw... .

My question is, is this article an objective, factual representation of the 9/11 community to legal officials? Unless the information presented to Mr. Spitzer is documented significantly beyond the contents of this article, then we are in trouble. If this case gets tossed aside as a representation of the 9/11 community’s best work then what happens when solid information has an opportunity to make it to the top in the future?

It is always best to explain yourself when providing constructive feedback. My name is Russell Pickering and I recently launched a website looking into the incident at the Pentagon. My hope is to push all of the evidence to the front (no matter what it is) and challenge ALL of our individual personal theories towards a comprehensive factual accounting of how they pulled off 9/11. I have put my time into matters related to the Pentagon investigation and care very much about the truth being known someday. I believe this will happen if we all fairly critique and support each other.

I exchanged many emails with Karl and sent him photos and ideas regarding his A-3 theory. After many hours of looking into it I abandoned the theory. I will point out a few observations on Karl’s article and some of my own opinions on the A-3 theory.

The article to "prove" a physical evidence case starts off with rhetoric and generalizations about the Bush family. George Bush in my thinking did not design and carry out 9/11. It was way beyond his scope and his intelligence level. I strongly dislike our appointed president but don’t feel he decided on the details of 9/11 operations although I believe he was aware of it. I feel this was irrelevant in making an objective case for an A-3 Skywarrior hitting the Pentagon to legal officials. Worse yet is that by the time they get to the A-3 theory they may already have shut their minds.

Karl says, “That has not been easy and we have had many working on this night and day for three years to get to the bottom of the matter.”

I don’t see three years of research represented in Karl's article in any way. There are no references for even the source of the images. He also says, “Our team had to take steps to go around the content blocks to get at the photos you are seeing regarding these rotor hub components.”

I found all of those photos in one night on Google. There is nothing original there. They are all clipped from various websites. We do not see any new photos, definitive part numbers or references that Elliot Spitzer or any of us can verify.

There is truth mixed into this article no doubt. Some things that we all pretty much agree on are the undisclosed videos from the Pentagon, the bogus war on terror, and the lack of credibility regarding the 9/11 Commission. But to alternate from things that we agree with and that evoke an emotional response back to a weak theory is not going to convince an attorney general. We should also keep in mind that an attorney general on the official side of the story might actually find those statements alienating - limiting their response to any facts that might be present. If you sent a plain and well documented case with substantial evidence that an aircraft other than Flight 77 hit the Pentagon then you might raise a brow.

Karl says, “It is not a “turbofan” component, it is in fact a “turbojet” component from an US Air Force/Navy vintage type of jet engine technology that was used on just a limited number of fighters, bombers and reconnaissance planes.”

Here is Pratt & Whitney's list of what the JT8D was used on http://www.pratt-whitney.com/prod_mil_jt8d-219.asp .

Go to Google and type in "jt8d turbofan" and then "jt8d turbojet". All experts say that the JT8D is a turbofan except for Karl (see statement above). When you enter "jt8d turbojet" it is only references to Karl's article. In three years of research it seems that Karl would have realized this http://www.shanaberger.com/engines/JT8D.htm . Even the diagram further down on this page labels it as a turbofan. Beside that point, the difference between these two types of engines is described well here, “A further variation on the turbojet is the turbofan. Although most components remain the same, the turbofan introduces a fan section in front of the compressors.” http://www.aerospaceweb.org/question/propulsion/q0033.s...

Karl says, “There was one credible witness found that saw "a two-engine jet airplane, the engines were under the wings." That is a visual description of a 737, 757, or 767, but it is also a description of an A-3 Skywarrior.”

I didn’t know that only one witness was considered “credible”. The eyewitness reports were all over the map, everything from a turboprop to a commuter plane including a 737, 747, and a 757. But not one described a fighter type jet that I am aware of (correct me if I’m wrong). There were sounds reported like that of a missile or a fighter but no visuals. One radar controller commented on the maneuver of the aircraft being fighter-like, but not with direct visual contact. Let’s just say it could have been a fighter jet, but was it an A-3? http://www.pentagonresearch.com/eye.html

Karl says, “It was very difficult to find the exact FAA certified company that is equipped, tooled or certified to work on the jet engines that were used in the A-3 Skywarrior.”

I challenge this. Just put "jt8d repair" into Google and you will find plenty of shops certified to do the work. It doesn't take secret research to find JT8D repair.

Karl says, “For those of you that do not pay attention to such things, 9-11 Commission member Jamie Gorelick sits on the board of directors of United Technologies and Pratt & Whitney is now owned by United Technologies. Former Bush EPA secretary Christine Todd Whitman also sits on the United Technologies board, and that is a company that is raking in big bucks due to Bush War Policy.”

What does this have to do with anything? Because these people were on the board they had some factor in choosing the A-3?

Karl says, “The FEMA photo shows such a front seal and a rotor hub missing its fan blades. The key is the cleats on the outer perimeter that the fan blades attach to and these particular blades are easy to knock off in a collision such as that at the Pentagon.”

Those cleats can have any size of fan blade put in them or none at all. That component is similar to many engine parts in many various engines. See the diagram below of the JT8D turbofan to see the different size blades in the front..

Karl says, “Some of the team kept digging and they finally found what might be the actual type of part that was at the Pentagon. The part that is visible in the left of this photo could well be the exact part, and I can assure all that what you are looking at is not the front fan of a 757 jet engine. That is a 737 type of "turbojet" power plant and might well be in the "modified A-3 Skywarriors." I bet that jet mechanic from Evergreen Air (a known CIA affiliated company) can tell us where that fan blade comes from and do so in front of a Spitzer or Morganthau grand jury.”

Maybe based on this photo from the internet they will subpoena that mechanic from the “CIA” shop and get him to confess?

Karl says, “What has been interesting is the level of "content blocking" that there is on the Internet where specific information regarding certain "jet engine components" such as those shown at the Pentagon have definitely been blocked. Our team had to take steps to go around the content blocks to get at the photos you are seeing regarding these rotor hub components.” Again he says, “…..it took my team over two years to hammer through such blocks to find three of these photos (Praxair and Evergreen) to verify the component.”

One more time - I found them and many more in one night on Google. I sent Karl many JT8D photos and diagrams of internal parts. I don’t know why he would exaggerate this to sound like the whole internet has been scrubbed of any information regarding these parts. Now I agree that reference and documentation links are being removed from the internet regarding 9/11. I have seen that with my own eyes and that is why I save every website I find that has information I need.
Karl says, “Yes, Hughes aircraft had a fleet of them and was bought out by Raytheon. Hmm, that company is doing well for two reasons that I know of due to Bush war policy and even the move from Mode 4 to Mode 5 technology since the PRC got its hands on our top secret Mode 4 technology with that little Hainan Island incident and our Navy EP-3 that was forced down in April of 2001. As of Sept. 11, 2001, most air traffic controllers and National Air Guard units were not upgraded after the PRC got their hands on some of our most sensitive military technology.”

More top secret stuff. Is that going to convince an attorney general on the behalf of the 9/11 community that Flight 77 didn’t hit the Pentagon?

Karl says, “The above photo is a launch of an AIM-54 Phoenix Missile, air-to-air missile. Making this launch an air-to-surface missile would not be a great feat and Hughes/Raytheon manufactures several such missiles that would blow a 16-foot diameter hole in the Pentagon with ease.”

I have two problems with missile theories in general. One is that there were many motorists right on the scene that did not report a missile of any sort (they used analogies but that was it). One thing we forget is how many people were there that didn’t give a report at all. They just drove away. There are certain types of people who make sure they get their account heard. These people for various reasons come forward and that is who we heard from. I’ll bet if a missile skimmed over the highway with a fighter jet right behind it we would have heard from a lot more people in that case. Let’s remember there were firefighters on the ground, tower personnel in the helipad tower, and people in all of the fully occupied offices to the North of the renovation zone with windows looking right out onto the lawn. We probably didn’t hear from the majority of them. But if a jet fighter came in and launched a missile I’ll bet we would have. The other problem with the missile theory is what about the exit hole? How did a missile blow a near perfect 9’ diameter hole as clean as that 310 feet into the building and then decelerate in the distance of AE Drive to not even chip the opposing wall? That is where I tend to side with theories that suggest the use of secondary explosives in the building. Also, does the AIM-14 have the ability to travel 310 feet through a building including and at least 6 feet of concrete inner walls, 5 feet of steel reinforced concrete exterior walls, 16" of brick, 6" of limestone, all of the interior pillars and the office contents (this I don't technically know but it would be interesting if it could since even the specially designed bunker buster is rated for only 20' of concrete http://science.howstuffworks.com/bunker-buster2.htm ).

Karl says, “Also note that jet engine fuel burning does not leave a white trail at sea level, only at higher and much colder altitudes. However, AGM type missiles burn solid rocket fuel and it leaves a white trail at all altitudes.”

This is very true. In the 5 video frames released it does have the white trail. Also, if there was nothing to hide then why release the frame where the aircraft is hidden? But if the frames showed a missile trail why release them and expose that? Those are good unanswered questions. One possibility is that an engine took in a piece of debris from the lamps and stalled the compressor spraying fuel vapor. I would like to hear somebody address this possibility in detail since I have not found a good answer. One common theme in the eyewitness reports is the engines “revving” up after hitting the poles. As somebody who was an aircraft mechanic for 5 years 9 months in the USAF I do know that engines make funny sounds after ingesting something from the training films that I watched. Many eyewitnesses reported a change in the engine sound after hitting the poles. http://www.pentagonresearch.com/023.html

Karl says, “However, the following is the diffuser case design for the 757 jet engines and it is quite different from that shown at the Pentagon. That is due to the difference between "dual-chamber turbojet" versus the newer "high bypass jet fan" designs found on the 757 and 767 jet airplanes.”

He does not show the Boeing RB-211 diagram that is well known on 9/11 research sites and that I sent him personally. The part at the Pentagon is very similar to that diagram. It is not a GE part as in his diagram. It is also not a "diffuser", it is the combustion chamber. If Karl would like to post the reference to the “diffuser case” photos that he shows with a part number or something that we can verify, I would like to look into those further. http://www.pentagonresearch.com/081.html

Karl says, “They also made the wheels for the 757 but a simple proportional check of width versus diameter will easily show that the below photo is not of a wheel hub from a 757, which has a much larger radius than width. This radius being about the same as the width of the wheel hub is also another clue that the 757 story is a Bush Lie. In fact, if one looks very closely at the diameter versus width of the tire that was found at the Pentagon, this is the type of tire used for carrier based and general rear wheels of smaller military planes, not commercial airliners. This is the type of wheel hub one would expect to find as one of the two rear wheels on an A-3 refitted with current equipment rather than equipment that is no longer being manufactured.”

The right photo above is not from the Pentagon. It is from the WTC.

I would like to see the proportional “check” numbers and how that was accomplished from a photo compared to an A-3. I’m sure the attorney general would too. The idea that a special wheel was made to “refit” the A-3 for this attack is doubtful. I have looked at a ton of A-3 photos (some of them recent) and they all have the same rim (See below). Rims on aircraft are integral to the landing gear including brake and hydraulic fixtures. There are not just interchangeable.

While I was researching Karl’s theory, the A-3 did give me one hope. I thought maybe its tail wheel bumper was the mystery part in the Pentagon that was being attributed to a landing gear. There was only one of them and it had similarities. (See below)

But after contacting a couple of old A-3 guys and finding some other photos, I found this to not be the case. On my website that piece in the Pentagon does turn out to be a 757 main gear. Go to http://www.pentagonresearch.com/757debris.html and click on the photo of the piece to see another photo of it, the part numbers and the history.

The thing yet to be proven is the use of the JT8D on the A-3. It is usually listed as the J-57-P-10 http://www.globalaircraft.org/planes/a-3_skywarrior.pl . Put "skywarrior jt8d" into Google and again the only references to this are related to Karl's article. No aircraft experts list the JT8D in use on the A-3.

My conclusion is that Karl’s presentation is not an accurate representation for someone at the level of an attorney general who may dismiss future 9/11 research based on it. It does not provide part numbers or the opportunity for external verification or even sources for images. Most of all, it will not produce objective credibility in the minds of those already skeptical of people questioning the official story. It does not deal with additional debris at the scene and numerous other established issues. I will correct myself if found to be wrong.

Any theory that ends up being very close to the truth will have to account for all of the witnesses, building anomalies, evidence (planted or not), and the actual things we know to be true about the incident. The 9/11 operation goes so far beyond all of the smaller stuff we sit around and argue about. We have to get past our pride and really start to look for what matters. The option is to sit and watch the History Channel in forty years chatting about it as a possible “conspiracy theory”.

Russell Pickering
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k-robjoe Donating Member (1000+ posts) Send PM | Profile | Ignore Sat Mar-12-05 09:18 AM
Response to Reply #32
49. sound barrier
> "The sound barrier was broken that morning and witnessed by an entire school yard and many residents."

Havent heard about this. Any links?
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Christophera Donating Member (1000+ posts) Send PM | Profile | Ignore Wed Mar-16-05 01:52 AM
Response to Reply #49
51. Links To Sonic Boom Witnesses
dragonslair site is no longer but the other link is good.

http://www.dragonslair.pwp.blueyonder.co.uk/77/ffdd.htm...
Rob Schickler, a Baylor University 2001 graduate and Arlington, Va. resident, said.
"A plane flew over my house," (one mile away from the Pentagon).
"It was loud, but not unusual because the is by my house, on the other side of the Pentagon. Occasionally planes that miss the landing fly over my house."
"A few seconds later, there was this sonic boom," he said. "The house shook, the windows were vibrating."
"There was a hole in the building, and you could smell it in the air. It's a beautiful day, but you can smell the burning concrete and burning jet fuel."


http://news.nationalgeographic.com/news/2001/12/1211_wi...
Hortum rushed to the church school, filled with kids. "We're under attack, under attack," he whispered to Christine Yeannakis, the school director.

"I figured he was telling me the vestry was upset about something. We had no idea," Yeannakis recalled.

Then a supersonic boom from somewhere in the sky hushed the playground and Yeannakis hauled the kids into the basement.

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mikelewis Donating Member (1000+ posts) Send PM | Profile | Ignore Wed Mar-16-05 08:52 AM
Response to Reply #51
53. That could have been the sound of the plane exploding into the building
I don't believe it was Flight 77 that hit the pentagon but that's not conclusive.

Bt here's something I find interesting

"It was another 20 or 30 minutes before there was anything on the television that anything had even happened in Washington. This was the first word I got," he said."

There were reports of all sorts of unconfirmed activity in Washington, why would it take so long for the news to report it. 20-30 minutes is a long time. Is this true or just this guys perception?
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Lithos Donating Member (1000+ posts) Send PM | Profile | Ignore Fri Mar-11-05 10:14 PM
Response to Original message
46. You have to use FEA (Finite Element Analysis)
To even get a reasonable approximation. Any attempt to to this by "hand" requires so many assumptions as to be unrealistic.

L-
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