WTC Pancaking and Conservation of Momentum: Establishing a Freefall Baseline

I was arguing with someone about this idea that the World Trade Center towers fell at freefall speed. I pointed out that if you look at this clip of WTC2, it’s pretty easy to count the seconds. It begins collapsing around second 10, and you can see the blowback from the top section hitting the ground at around 28 seconds, so about 18 seconds to fall. Since the bottom of the top section is around 300 meters off the ground, freefall time would be sqrt(2.0*300/9.8), or about 7.8 seconds.

So then he says he’s got this Matlab model that applies conservation of momentum to the pancaking and gets a freefall time of 15 seconds, indicating that there’s hardly any resistance to the collapse, and isn’t that fishy.

In response, I made my own conservation of momentum model of pancaking. Okay, what’s that?

WTC2 was hit at around the 78th floor and above. When it began to collapse, it was that 32 floor top section that started falling first. It hit the 77th floor and gained the mass from that, then it hit the 76th floor and gained the mass from that, and so on and so forth. Each time it gained mass, it had to slow down a little because momentum is always conserved. In solving problems like these, it’s good to start with a simple model first.

The top section fell 3.77 meters, the distance between floors, and gained sqrt(2.0*9.8*3.77) or 8.6 meters per second of speed. In this simple model, it then collided inelastically with the 77th floor, and lost 8.6*(32/33) or 0.26 m/s, and then started accelerating again. Each time it hit a floor, it lost a fraction of its velocity, but then gained more back as it fell. The fraction at the next floor would be 33/34, next would be 34/35, and so on until the last fraction would be 108/109.

Over the entire collapse, this freefall pancaking conservation would add 1.76 seconds to the time of collapse, for a total collapse time of 9.56 seconds, yet the actual fall took 18 seconds. Clearly this is much slower than freefall, there is some resistance mechanism, but the nature of that mechanism is beyond the scope of this article. Note that the actual fall time is about twice the freefall pancaking model time.

A similar analysis of WTC1, yields similar results. You can see from this clip of WTC1 collapsing that it takes about 23 seconds to fall. It was hit at a higher floor, the freefall pancaking model takes about 11.5 seconds. Again, actual fall time is about twice the model fall time.

Here’s the little program I used to do the calculations.

<?php
$height = 415.0; // Height of WTC2 in meters
$n = 110.0; // Number of floors
$s = $height/$n; // Distance between floors
$a = 9.8; // Acceleration of gravity in meters per second per second
$h = 78; // Floor hit by plane
$floors = 110-$h; // Floors to fall
// Velocity of the falling mass after one floor
$vm = sqrt(2.0*$a*$s);
// Time for first floor to floor drop
$dt = sqrt(2.0*$s/$a);
// Total time
$t = 0;
// Initial distance fallen
$x = $s;
echo "<table>";
for($i = 0; $i <= $h; $i++)
{
// Add the time to fall to the total time
$t += $dt;
echo "<tr>";
echo "<td><pre>$floors</pre></td>";
echo "<td><pre>$vm</pre></td>";
echo "<td><pre>$dt</pre></td>";
echo "<td><pre>$t</pre></td>";
echo "<td><pre>$x</pre></td>";
echo "</tr>";
// Velocity after colliding with the next floor down
$vc = $vm*($floors/($floors + 1));
// New velocity of falling mass before next collision
$vm = sqrt($vc*$vc + 2.0*$a*$s);
// Time to drop one floor
$dt = 2.0*$s/($vc+$vm);
// Calculate the distance fallen to double check the model
$x += $vc*$dt + ($a*$dt*$dt)/2.0;
// Accumulate mass
$floors ++;
}
echo "</table>";
?>