Calculation of World Trade Center Building 7 Collapse Acceleration

Accurate measurement of the total time for the global collapse of World Trade Center Building 7 is difficult or impossible due to the fact that little footage exists showing the collapse, and of the videos that are available, none provide both a clear view of the entire height of the building and show the complete collapse from beginning to end. Surrounding buildings as well as voluminous smoke and dust clouds from the collapse itself obscured the base of the structure, preventing accurate estimations of precisely when the collapse finished. However, video does exist with identifiable and measurable height reference points to enable a calculation of the fall acceleration rate to a reasonable level of accuracy.

The video I will use was recorded by CBS news and can be downloaded here: VIDEO LINK

Figure 1 is a captured still from the video, showing the North face of WTC7.

The white building with stepped upper levels in the right mid-ground is 101 Barclay Street, also known as the Irving Trust Operations Center. In the CBS video, the top right (northwest) corner of WTC7 is visible as it passes the first (lowest) step on the Barclay Street building. With the height of WTC7 and the height of the Irving Trust building known, we can determine the distance between the top corner of WTC7 and the first step on the Irving Trust building, then using the time taken for the collapse to cover that distance calculate an average acceleration across that distance.

The intent is to measure the average acceleration of the collapse of the main structure as opposed to the total collapse time of the entire building, hence the separate collapses of the penthouses on top of the building are not considered as reference points. The top right corner will be used as the reference point, with time = 0 and velocity = 0 being one frame before the first pixel of downward movement of the corner, and total time being when the corner is level with the first step of the Irving Trust building.

101 Barclay Street Height Information on the Irving Trust building can be found here:

http://www.emporis.com/en/wm/bu/?id=114274

http://www.wirednewyork.com/101_barclay.htm

Both references state the building height as 99 meters. The number of floors above ground is different from both sources, one stating 23 and the other stating 25, however Figure 5 shows 26 floors, which is the figure I will use.

There were seven stepped floors on the upper storeys, with the first step comprising the 19th floor. With the height across all the floors being even, the height of the 19th floor is (19/26) x 99 = 72.346m

Building 7 Height

Considering NIST's extensive study of the building it can be reasonably assumed that their data is more accurate, however both heights will be used in two separate calculations. The penthouse is assumed to be included in the stated heights, thus providing a conservative estimate of the height of the northwest corner. I will assume an arbitrary height of 10 feet (3.048m) for the height of the penthouse since data on this appears unavailable.

Therefore, the reference vertical distance is the height of WTC7, less the height of the penthouse, less the height of the 19th floor of the Irving building.

NIST: y = 185.928 - 3.048 - 72.346

= 110.534m

Emporis.com: y = 174 - 3.048 - 72.346

= 98.606m

Time Measurement The first downward movement of the WTC7 northwest corner occurs just after the 8.343 second mark.

The corner is level with the top of the 19th floor of the Irving building at 13.100 seconds.

So, t = 13.100 - 8.343 = 4.757s

Average Acceleration Calculation For a falling object with an initial velocity of zero: y = 0.5at2 => a = 2y/t2 NIST: a = 2*110.534/4.7572 = 9.77 m/s2 Emporis.com: a = 2*98.606/4.7572 = 8.71 m/s2

Conclusion

I estimate an error margin of +/-2m for y, which would translate to +/-0.175 m/s2 for the final acceleration figures. Accuracy beyond this exercise would require much clearer footage and, of course, confirmation of the correct height of the northwest corner of Building 7. Should the reader have any information in this regard, please do not hesitate to contact me at winston@studyof911.com

Addendum

9.8m/s2 is the rate of gravitational acceleration in a vacuum, and the rate of collapse of WTC7 is often compared to this. As an exercise, we can roughly determine the expected air resistance experienced by the building during collapse and further take that into consideration. The method I use is a simplified air resistance calculation method for a simplified model and will produce a rough idea of the expected reduction in fall acceleration if the building were in free fall in atmosphere as compared to in vacuum. Calculating air resistance to a high degree of accuracy would be not just prohibitively complex, but simply unnecessary in this context as the negligible amount calculated below will show and as would be expected for such a massive structure.

Drag force on an object is given by the equation

Fd = 0.5*C d *p*A*v2

= bv2

b is a constant where,

C d is the drag-coefficient of the object, a dimensionless constant dependent on its shape and determined empirically. The air in the individual floors of the building would move with and can be considered one unit with the structure and therefore ignored in the calculation. Only the bottom-most floor at any one point on the collapse would experience air resistance as it must push air out of the way in order to impact the ground. The drag co-efficient for a long flat plate is 1.98[1] and will be used in this case for simplicity sake.

p is the density of air (assumed near sea level) 1.223Kg/m3

A is the area of the object normal to flow. NIST cites the area of each floor as 2,000,000 ft2 = 185 806.08m2

Therefore,

b = 0.5 * 1.98 * 1.223 * 185 806.08 = 224968.4274816

An object reaches terminal velocity when the drag force acting on the object in the negative equals the force applied to it in the positive, in this case gravity.

Hence at terminal velocity Fd = bv2 = mg

=> V T = SQRT(mg/b)

No figures are available for the mass of WTC7 that I can find, so I will make a very rough estimate based on other published estimates of the mass of WTC1 (5 x 108 Kg). WTC7 was 0.45 times the height of WTC1 and covered comparable floor area, however since there was less steel used in the construction of WTC7, I'll use an arbitrary figure of about a third, placing the mass of WTC7 at 1.7 x 108 Kg

V T

= SQRT(1.7 x 108 * 9.8 / 224968.4274816)

= 86.0551195 m/s

Velocity at time (t) is given by v(t) = V T tanh (gt/V T )

Velocity at time 4.757s

= 86.0551195 * tanh(9.8*4.757/86.0551195)

= 42.53670521674826m/s

Average acceleration across this time

= v/t

= 42.53670521674826/4.757

= 8.94m/s2

The final figure for theoretical collapse acceleration rate of WTC7 in complete free fall in atmosphere and at sea level is 8.94m/s2, which is only a little above the actual observed 8.71m/s2 acceleration rate arrived at from analysis of the CBS footage and using the Emporis height measurement. From this we can imply that the structure provided a negative acceleration, i.e resistive force of approximately 0.23m/s2 to the gravitational collapse.