Reply #178: Loss in potential energy...

[Carefulplease]

I seem to have misread this the first time.

Here's how I got 20%: I calculated how long it would take the buildings to collapse in near free fall. By "near free fall," I mean the scenario that seems to me most consistent with the pancake theory: everything above the point of impact falls to the next floor, which is sitting unsupported but unmoving, waiting to be hit. Then gravity plus the momentum of the moving mass cause that floor to slam into the next one, etc, all the way down. I assumed all the energy of motion went into accelerating the next floor down. Realistically, of course, you would not expect a perfectly elastic collision, but I figured this was a good baseline assumption for later comparison. It came out to 9.2 seconds for the tower that was hit on the 85th floor. If the actual time of collapse was 10 seconds, that means the tower's average speed of collapse was (rounding off) 9/10 of what it would have been unimpeded. So (squaring and rounding off) its kinetic energy was 8/10 of what it would have been in freefall.

That 80% became kinetic energy and brought the towers to the ground. What happened to it after that I don't know.

If you assume perfectly elastic collisions then it also gets rather complicated and unrealistic. The top section comprising 15 stroreys (F85-F110) falls as a block until it hits the next floor below (F84) with speed v1. After this first elastic collision this floor moves with an initial velocity that is close to 2*v1 while the velocity of the upper block is reduced slightly. This first impacted floor then accelerates further under the action of gravity until it hits the next floor (F83) with some speed v2 > v1. After the second elastic collision, the velocity of floor F84 is reduced to zero and the velocity of floor F83 becomes 2*v2. The upper block is thus allowed to keep up with F85 and hit it one second time. Thereafter the sequence of collisions become very complicated and chaotic.

The assumption that collisions are elastic amounts to the presupposition that all the energy from the impact is converted into elastic strain energy and fully restored such that the relative velocities of the two impacting masses are reversed. It seems to me to be more realistic to assume that the energy is mostly absorbed in the reduction of the relative velocities through expelling air and producing plastic strain and fractures in the floor trusses, concrete and live load materials. Only some fragments will rebound, not the whole floors.

I would also question your assumption that the collapse duration is 10 seconds. Refer to Greening's paper for more reasonable assumptions and measurements based on video evidence.

On edit: have a look at message #15 in this thread for more references on the collapse times.