did they come down too fast?
Here, I wish to deal with the fall rate in the context of conservation of linear momentum ... one of the cornerstones of physics. I have heard of physicists doing the calculations for this particular fall and getting as much as 40 seconds (as a minimum fall time) when including conservation of energy considerations. But that is a more difficult estimate because I don't know what weight each floor was designed to take and what distance to apply that force through ... to get an energy number. So I am sticking to linear momentum conservation alone which is the easier calculation because it is independent of the mass of the floors.
If there is a possible way to get around linear momentum, it certainly wouldn't occur in a common fall and collision. Understand here that there is no known instance of a violation of conservation of linear momentum in the entire earth's scientific experience. So, if one shows that linear momentum conservation is violated in the fall of the towers ... then ... they were brought down by demolition ... no ifs ands or buts.
The calculation is simple enough to be understood by anyone who has taken high school algebra and physics. I'm going to do it the long way (floor by floor) rather than shortening the problem into a single integral equation. Be patient.
Let's make the equations first
The distance an object falls in the earth's gravitational field at the surface (ingnoring air friction which slows the fall) is ...
Where "a" is the acceleration due to gravity and is 32 feet per second per second ... 1/2 a = ~16
The velocity an object achieves after falling a distance (D) is simply ...
And, the time of the fall is ...
Let's check it out ...
The tower is 1360 feet tall, so the time of free fall ... if the tower's mass was all hovering at the 1360 foot level in one thin block and suddenly fell in a vacuum ... would be ...
the square root of 1360 divided by 4 = 9.2 seconds (this is the absolute minimum fall time for the towers if demolition charges were expertly placed and the building came down without any resistance whatsoever)
So, if something falls in 10 seconds even, it must have started its no-resistance fall at ...
16 times 102 = 1600 feet up in the air ... and so on for any other height.
Now consider a two-piece building
Here, half of the mass of the building is compressed into a flat piece at 1360 feet ... and the other half of the buildings mass is half the distance down in a similar flat piece. Let the top piece free-fall and slam into the other half at the 1/2 x 1360 = 680 foot level.
We can compute the velocity of the top piece when it hits the second piece. We need the time of fall first which is ...
6801/2 / 4 = 6.52 seconds
Then, the velocity is ...
6.52 x 32 = 208.64 feet per second (about 142 miles per hour)
Now, when the top piece hits the stationary piece at 208 feet per second it will be slowed and the stationary piece propelled down with it. They go now as one piece and the law of conservation of linear momentum requires that the new velocity be half that of the one piece by itself. Thus, we have one piece at 208 fps and another piece at 0 fps = two pieces pancaked together at 104 feet per second. Then we add to that velocity ... whatever additional velocity that as the now doubled mass will acquire by gravity through the remaining 680 feet to ground level.
In general, if we divide up the building into such equal pieces (floors) and let them "pancake" into one another progressively ... the slowing of velocity due to collisions of stationary floors with the "falling-as-one floors" ... will be inversely proportional to the mass increase. Thus, if 6 floors are falling as one and collide with yet another stationary floor, their mass will then increase by 7/6 and their velocity will be 6/7 of the collision velocity. If the increase of mass is 9/8, the decrease in velocity will be 8/9, and so on.
Linear momentum is then conserved since 6 masses with velocity "1" collide with 1 mass of velocity "0" and we end with 7 masses falling-as-one with a velocity of 6/7. Thus,
Adding up the velocities
We have to add the velocity acquired by acceleration due to gravity, to the initial velocity of a falling object. Thus, in addition to
agravity x ttime = vvelocity from continued acceleration
we have to add the "before collision velocity" in order to obtain the final velocity of collision. We do this by defining the distance between floors. Thus, if we make the distance between floors 680 feet as above we get ...
Which is a simple second order equation (quadratic).
0 = 16t2 + vit - 680
Now, we cheat and go to www.akiti.ca/Quad2Deg.html ... plunk in the coefficients and click on "Calculate the Solutions" and we get t (the time of fall from one level to another with both the new acceleration and the old initial velocity accounted for).
Here's a compressed screen shot of the web page to calculate from. Remember to make your distance between floors negative and that gives the physically appropriate solution.
Then we just do the same thing over and over and over for each floor ... then add up all the times between floors ... and that's the total time of collapse possible. Remember, this is absolute. There is no appeal to conspiracy or lack thereof. Nature doesn't care about us one way or another.
Now, let's do some examples
For one floor we already have 9.2 seconds.
For two floors we get this ...
Let's do it again for three floors ...
I did this for 4 floors then 5 floors then for ten floors and got total collapse times of ..
1 floors = 9.2 seconds 2 floors = 10.55 seconds 3 floors = 11.35 seconds 4 floors = 11.77 seconds 5 floors = 12.13 seconds ----- ----- ----- ----- 10 floors = 13.09 secondsSo you can see that the time of collapse gets longer as we divide up the mass of the building into more and more equally spaced floors of equal mass. Understand too that air resistance would make the fall take a tiny bit longer. Junk falling off the sides to free fall on its own would also slow the collapse because those masses would contribute to the slowing on impact but not to pushing the process forward through the other floors. Also note that the amount of mass has no effect on the velocities because everything ... no matter what its mass is ... falls at the same rate in the earth's gravitational field (as per Galileo).
Basically, the problem is that no matter how you arrange things, the top floors have to accelerate the lower floors and in turn are slowed so that a pancake collapse necessarily takes awhile longer than just a straight free fall. If ten floors fall, they must accelerate the next ten floors from their initial stationary position. Given what I've done already, I think a collapse of 15 to 16 seconds for 100 floors could be expected. Note here that though the addition of more floors slows the process ... they slow the process at an ever decreasing rate. There's probably a limit in there someplace.
My conclusion is that the fall of the towers is inconsistent with the pancake model. The floors beneath the point of initial collapse must have given way prior to the arrival of the top floors, i.e. by explosive demolition. Understand that I haven't attempted to include resistance of the steel in the towers to the collapse which would have further retarded the progress of the fall. With that in place a fall of 20 or more seconds is not unreasonable. 11 seconds is definitely unreasonable.
This is why the N.I.S.T. report does not follow the collapse past the point of initiation. If they did, they would run into contradiction with the fundamental laws of mechanics and the entire scientific community would have been sent howling.