What if...

...all the floor slabs of the Twins hovered in mid-air and needed only an infinitesimally small impulse to drop towards earth's center of mass, and no breaking, shearing, tearing, buckling, pulverization happened at all?

Intro: the Honda collision law.

The height of the South Tower is 1362 feet. I calculated that from that height, freefall in a vacuum (read, absolutely no resistance on earth) is 9.2 seconds. According to testimony provided to the 9-11 Commission, the tower fell in 10 seconds. Other data shows it took closer to 14 seconds. So the towers fell within 0.8-4.8 seconds of freefall in a vacuum. Just like WTC7, this speed seemed impossible if each of the 110 floors had to fail individually.

As I was considering this, another problem arose. There is a principle in physics called the Law of Conservation of Energy. There is also the Law of Conservation of Momentum. I’ll briefly explain how these principles work. Let’s assume there are two identical Honda Civics on the freeway. One is sitting in neutral at a standstill (0 mph). The other is coasting at 60 mph. The second Honda slams into the back of the first one. The first Honda will then instantaneously be going much faster than it was, and the second will instantaneously be going much slower than it was.

This is how the principle works in the horizontal direction, and it works the same in the vertical direction, with the added constant force of gravity added to it. Jim Hoffman, a professional scientist published in several peer-reviewed scientific journals, took a long look at all of this. He calculated that even if the structure itself offered no resistance, that is to say, even if the 110 floors of each tower were hovering in mid-air, the ”pancake” theory would still have taken a minimum of 15.5 seconds to reach the ground.

So, even if the building essentially didn’t exist, if it provided no resistance at all to the collapse, just the floors hitting each other and causing each other to decelerate would’ve taken 15.5 seconds to reach the ground.

This program computes total collapse times based on parameters describing the floor on which the collapse started. The program makes the following assumptions, all of which favor short collapse times.

• Each floor is an infinitely thin slab, and all the mass of a story is concentrated in the slab.

• Mass is uniformally distributed among the stories.

• The overhanging portion (eg: 14 floors in the North Tower) falls as a block, with its bottom floor accumulating pancaked slabs of the once-intact floors as it encounters them.

• Once the bottom of the block reaches the ground, the floors in it start to pancake from bottom to top, the roof of the tower falling at freefall at that point.

• The falling block remains perfectly centered over the intact portion.

• The accumulation of floors is inelastic.

• Each floor’s support vanishes when touched by the falling block.

• Momentum is conserved.

• None of the kinetic energy of the falling mass is diverted to other sinks (concrete pulverization, steel bending, etc.)

The following table summarizes the results of running the program with parameters specifying that the collapse starts at the 80th and 95th floors:

start floor	crash zone to ground	roof to ground
80	9.733 s	11.613 s
95	11.604 s	12.608 s

In 2006, Hoffman created a generalized version of the program that allows the removal of the last two assumptions. In particular, it allows the specification of:

• A linear increase in the mass of stories from the roof to the ground.

• The movement of a fixed fraction of the mass falling within the Tower's profile to move outside of the profile with the collapse of each story. Once mass moves outside of the Tower's profile, it does not participate in the acceleration of mass downward.

The Twin Towers, as all large steel-framed skyscrapers, had columns that became less massive at increasing elevation. This means that the Towers' upper stories were considerably lighter than the lower ones. The assumption that the lowest stories were about 1.5 times as massive as the top stories seems like a reasonable assumption. Implementing this adjustment to the model means longer collapse times, of course, since more of the mass would initially be lower in the Tower where it would have less mass underneath it to accelerate downward. However, the simulation shows that even making the mass ratio of the bottom to the top story 2.0 has relatively little effect on total collapse times.

In contrast, the second adjustment of the new model has a pronounced effect on total collapse times. Assuming that just 6 percent of the mass above the impact zone is ejected outside of the Tower's footprint for each story crushed results in a total collapse time of nearly 20 seconds, assuming the collapse started at the 95th floor.

start floor	mass of bottom story relative to top mass	dispersal per story	mass dispersed by end	crash zone to ground	roof to ground
80	1.5	0	0	9.98	11.92
95	1.5	0	0	11.86	12.91
80	1.5	0.03	0.67	11.56	14.07
95	1.5	0.03	0.68	14.05	15.49
80	1.5	0.06	0.82	13.57	16.53
95	1.5	0.06	0.82	16.59	18.34

> > https://www.math.wisc.edu/~robbin/angelic/911.pdf < <

> > http://911research.wtc7.net/wtc/models/ < <

^{/r/towerchallenge}