Billiard Balls and Momentum
(Dr.) Judy Wood's "Billiard Ball Example" is a strong proof that the WTC towers were brought down by abnormal means and were not gravity-driven "collapses."
Her basic point is that in a progressive collapse, it will take a fair amount of time for each floor to drop to the next and keep the collapse going.
Using an ideal model, where each floor drops at free-fall speed to the next, but then pulverizes as it breaks the next floor free, and then the next floor begins falling at free-fall speed until it reaches the next floor, etc, she calculates a minimum collapse time of 97 seconds.
In reality, this model under-estimates the time, as each floor is NOT going to drop through the columns of the floor below at free-fall speed in a vacuum, but should go slower. In addition, the energy required to pulverize each floor is far more than the available kinetic energy (KE) AND the energy required to break up each subsequent floor support is far more than the available kinetic energy (KE). (Please see references provided.)
Looking at the data, we take the conservative approach that a falling floor initiates the fall of the one below, while itself becoming pulverized. In other words, when one floor impacts another, the small amount of kinetic energy from the falling floor is consumed (a) by pulverizing the floor and (b) by breaking free the next floor. In reality, there isn't enough kinetic energy to do either.[Trumpman][Hoffman] But, for the sake of evaluating the "collapse" time, we'll assume there was. After all, millions of people believe they saw the buildings "collapse."
The big issue many people have, including me when I first saw her billiard ball piece, is MOMENTUM. Intuitively, it is natural to think that there should be some momentum transferred from floor to floor during a progressive collapse- and that should speed up collapse time. The natural reaction is to think of a huge weight gaining so much speed that is smashes down through all lower floors, in a "pile-driver" type of reaction.
If you read down to the appendices of the Billiard Ball Example piece, Wood explains the logic of how momentum shouldn't be transferred during a floor collapse, and how floors coming down and bashing the lower floor will lead to the upper floors pausing as they knock loose the lower floor. These upper floors will then have to start falling again, which is what slows the collapse.
Is Wood wrong about momentum transfer? While there may be something out there refuting her, I can't see any high-profile clear refutation of Wood's analysis- nothing turns up on Google, anyway. There are discussion board refutations of her model, such as here- but this is hardly a rigorous debunking. It is more of an appeal to "common sense". Such as, 'come on, you KNOW momentum has to be transferred during a collision.'
So, I thought I would deal with the MOMENTUM issue.
I wondered if I could model the collapses where there was some degree of momentum transfer at each floor collision.
To do this, we need to use the following standard equations:
v = at and v(2) = v(1) + at
where v = velocity in feet per seconds, a = acceleration in feet (f) per second (s) per second, t = time in seconds, v(1) is initial velocity and v(2) is final velocity over a period of time t at acceleration rate a.
For these calculations, a is the force of gravity, 32 feet per second per second.
Thus,
v (f/s) = 32(f/s/s) t(s)
and
v(2) = v(1) + 32(f/s/s) t(s)
Also we need this equation
x = 1/2 at^2,
where x = distance travelled = 1/2 time t at acceleration a (squared).
Simply plugging in the height of one WTC tower, 1360 feet, and a as 32 f/s^2, we can easily solve for t and get a free-fall time of 9.2 seconds.
We will also need this equation:
x(2) = x(1) + v(1)t + 1/2 at^2
which tells us the distance x(2) travelled for a certain amount of time and certain acceleration, when there is an initial volecity v(1) starting at distance x(1).
So how do we work MOMENTUM into this system?
Momentum is mass times velocity. Here we will assume mass is constant for the mass falling at a certain velocity, meaning momentum is essentially velocity.
We will assume that there is free-fall between floors (this fast falling time, that assumes no impedances, greatly favors the official story) and that each floor is 12 feet in height. Also, we will assume that when one floor reaches the lower floor, it's velocity is decreased by a set amount each time- but that some velocity (essentially "momentum") is transmitted between floors to speed up the subsequent fall time between floors.
We will assume in this model that the only energy cost for destroying a floor is the loss of some degree of velocity at each floor collision.
The first set of calculations will be assuming that 50% of velocity is transferred at each collision of floors.
For the first floor, we need to calculate how long it took to fall to calculate its final velocity.
This can be done easily for the first floor using
x = 1/2 at^2
where x = 12 f
giving 0.87 s
Plugging 0.87 s into v (f/s) = 32(f/s/s) t(s), we see that the velocity after falling one floor is 27.8 f/s.
Now it gets more complicated for the second floor.
First, we assume that after bashing against the lower floor, that the two floors are going 50% of the original velocity - in this case 13.9 f/s.
Now we need to solve for time it takes to fall the next twelve floors, and we need this equation which takes into account the velocity the floors are already going:
x(2) = x(1) + v(1)t + 1/2 at^2
In our situation, x(1) = 0 and x(2) = 12 f, v(1) = 13.9 f/s and 1/2 at^2 = 16 (f/s/s) times t squared.
In other words-
12 = 0 + 13.9t + 16t^2
Now how on earth do we solve that?
Actaully if we re-arrange it into 16t^2 + 13.9t - 12 = 0, it is a quadratic equation- and we can solve for t easily with a quadratic equation calculator.
Now we get a time of 0.53 s for the 2nd floor collapse at free-fall speed and 50% loss of momentum. Now we can calculate the final velocity after this floor falls 12 f -> 30.9 f/s.
For the next floor collapse, the floor will start moving at 50% of this speed, at 15.5 f/s.
Going through the iterations for each floor, what happens is that after a few floors, a sort of terminal velocity is reached, where there is 0.5 seconds for each floor to drop.
If we assume that there is free-fall between floors and a 50% loss of velocity (momentum) used up by each floor/structural column collapse, the minimal time for collapse is 55 seconds. This is FAR longer than the maximal observed collapse time of 15 seconds per tower.
We can extend these calculations and test what happens if there is free-fall between floors and each floor/structural column collapse only slows velocity (momentum) by 25%. The calculated time is 36 seconds. This still is FAR longer than the maximal observed collapse time of 15 seconds per tower.
Lastly, we can extend these calculations and test what happens if there is free-fall between floors and each floor/structural column collapse slows velocity (momentum) by a mere 10%. This means again, there is pure free-fall speed between floors (which is highly unlikely) and only a tiny loss of velocity (momentum) when a floor is destroyed by the mass falling from above (extremely unlikely).
In this case, the calculated time is 23 seconds. This time still is significantly longer than the maximal observed collapse time of 15 seconds per tower, even though it assumes the lower structure offered very little resistance to the falling mass.
One could go on, but it gets even more ridiculous. One could say the structure only slowed the falling mass by 1% at each floor collapse, and get close to the observed collapse times, but this defies physics and is an insult to engineers who design strong buildings.
The bottom line is that even IF you take momentum transfer into account, the observed collapse times are far too short.
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A somewhat different issue someone might raise is that a progressive collapse could gain momentum (mass times velocity) with time- as the upper floors pile up, producing more mass to fall on lower floors.
The answer to that is two-fold.
First, the lower framework of any tower is always much stronger at the bottom than at the top, as the lower structure needs to support the upper weight. Thus, accumulating mass should be balanced out by the increased strength of the lower floors.
Second, what we saw on 9/11 was the upper floors disintegrating and turning to dust, with concrete being pulverized at each floor- such that there really was relatively little of a "piledriver" effect.
Lastly, one reason I built a model of the WTC, was to try to see if I could induce any sort of progressive collapse and try to observe how it proceeded. Unfortunately, because of scaling issues, the model was too strong to undergo ANY collapse. But I still would love to see a physical model of a stable structure (i.e. not a tower of blocks) that undergoes a progressive collapse!
Addendums-
1) If my calculations are incorrect, someone please tell me. The calculations aren't hard but I still may have made a mistake somewhere.
2) The method I used for calculating times gives a very different result than if you assume that there is constant acceleration during the progressive collapse, but that the acceleration is lower due to the resistance of the building. For instance if you assume that gravity can only pull down at 16 f/s/s (rather than 32 f/s/s) because the lower building structure offers even resistance, you get a fall time of 13 seconds for 1360 feet. But in real life, there would never be such even resistance- rather there would be a significant slowing at each floor.
3) In real life, if we assume a progressive collapse occurred for the towers, it would probably affect the calculations this way:
a) fall time between floors would be slower than free-fall speed due to column resistance and walls and various building components
b) the first collision would probably consume all or most of whatever momentum there was from the dropping floor- it is not clear at all that the first floor collapse would even break down the next floor to continue the collapse- but let's say the collision consumes 90% of the momentum of the top block of floors. If the next floor could then break away from the collision, then one might retain somewhat more momentum from the added weight, so there might be less and less momentum consumed at each floor collision. Thus one could try to model the timings with increasing momentum at each collapse. But overall, the timing would certainly well above the observed collapse times.
4) As far as I can tell from NIST, their explanation for the rapid collapse times is:
BUILDING HEAVY, GRAVITY STRONG. They have NOT offered any other explanation for the rapid collapses.
5) If we assume, by the official story, that a 15 story top chunk of tower (for WTC1) broke away from the lower structure due to a combination of plane damage, fire weakening and subsequent column stress, it is almost impossible that this chunk of building would gain significant momentum to break through the undamaged lower structure. The upper section of building is not going to suddenly break away and drop all at once at free-fall speed. Rather, it would start by the fire causing sagging on one side and the undamaged columns should take up the stress and eventually buckle and fail completely. But this process is not going to cause the upper section of building to plop down at free-fall speed- which is really what is needed to gain enough momentum for a progressive collapse.
6) In reality, for asymmetric damage there should have been more of an asymmetric collapse. Witness, what happened to WTC5 and WTC6, which underwent severe asymmetric collapse without undergoing global collapse. For WTC1 or WTC2, there should have been more of a local evidence of collapse before the global collapse was initiated.
7) In terms of 9/11 reality, of course the towers exploded into dense clouds of dust, the bulk of the concrete was pulverized, the whole structure fell apart at near free-fall speeds, and the debris pile was remarkably small– all signs of high-energy demolition.
In terms of academic analysis, the huge height of a WTC tower with 110 floors really makes it very attractive for mathematical modeling, and there are a lot of different approaches one could try. While mathematical models are nice to have, physical models are in fact the most useful and informative if done properly.