Computation of the g-forces involves a calculation based on the speed and the radius of curvature of the path. To determine the gentlest possible turns we used Geogebra, which is a free, open source, geometry software tool. We devised a geometrical construction such that given an approach line, a turn-off point on the line, a way point (labeled North Citgo in the diagrams), a target point, and duration for the roll maneuver, we could produce left and right arcs with equal radii and a straight segment between them, mutually tangent to the two arcs, representing the roll maneuver. Fig. 3 illustrates a typical result.
For a turn-off point level with Morin, with a 1 second roll time the results are:
Bank Angle (deg)
The design limit for a Boeing 757 is 2.5g. Even if the plane somehow held together, it would be impossible to control during such an extreme maneuver.
If we reduce the roll time to (a clearly fictitious) one half second the values improve somewhat, but not enough to bring them within the range of plausibility:
Bank Angle (deg)
Note that this scenario already conflicts with the testimony of Morin who stated that he watched the plane fly a considerable distance and descend behind a row of trees, with only the tall tail fin eventually being visible. It can be seen from the diagram (Fig. 3) that the plane would pass too quickly out of his line of sight. Note also that in this case the plane would disappear left wing first, his view obstructed by the vertical wall of the Annex, whereas we recall that he describes the plane disappearing from the bottom up, so his view must have been obstructed horizontally by the trees. We also see that witnesses would have seen the plane change from a steep left bank to a steep right bank between the Annex and the Citgo service station, but no such maneuver was reported.
If we now discount Morin’s description of the path of the plane entirely, and allow that he did not see it at all after it passed him, we can construct a scenario more favorable to the NOC hypothesis by moving the turn-off point earlier (Fig. 4). The last radar position provides the earliest point that the turn can reasonably be commenced. Morin’s observation that the plane flew nearly overhead is found to be preserved. Note that the curves are gentler.
Results, with a presumed 1 second roll interval:
Bank Angle (deg)
With a half second roll interval we have:
Bank Angle (deg)
This scenario, which totally disregards the testimony of Morin regarding the path of the plane (and there appears to be no justification for doing so) shows a substantially reduced g-force. It is still so high, however, that only someone with the skills of a trained fighter pilot would have a chance of performing it. The bank angle is still extremely steep.
The extraordinary bank angle
Anlauf and Paik would have seen the plane in a steep left bank and Morin would have seen it in a steep right bank. Hemphill would have seen the plane crossing from right to left of his line of sight to the impact point, at a steep right bank, as he looked out of his office window (Figs. 3 and 4 show his vantage point). These people reported no such thing. Hemphill repeatedly asserted that the plane was on his right and flying straight, and therefore with no significant bank. In the FDR file the maximum bank briefly recorded during this period was just 6°.
For these scenarios to work, the plane, after the roll, must maintain the steep right bank all the way to the Pentagon if it is to reach its target, which means the fuselage would have had to clear the roof of the Pentagon by nearly a wing-length, further straining the credibility of the “magic show” hypothesis.
The bank angle in all of these runs is so far out of the range of normal that, if it had happened, it would have astonished all observers. It would have been widely reported, yet nobody reported more than a slight bank. Albert Hemphill described the plane so close to the ground that he speculated about ground effect, which is clearly inconsistent with any of the calculated bank angles. Several of the witnesses indicated that the plane was flying “flat” in the vicinity of the Navy Annex, hence flying straight. This is totally at odds with the necessary curve and bank angle.
CIT has provided assistance here, handing some witnesses a model plane so that they could illustrate the bank. The bank they show is slight. In particular we note that not one of the 13 witnesses, who claimed they saw the plane well enough to believe that it was NOC, mentioned that it was extremely steeply banked. The bank angle would have been glaringly obvious and, because of its strangeness, unforgettable.
Notes on the Calculations
The images used for these constructions are from Google Earth, with the history rolled back to September 12, 2001, (or September 13 for those of us west of the International Date Line). Note that, due to the camera location for this particular photograph, the roofs of the buildings are displaced a little south east relative to the footprints at ground level. The undulation of the landscape can induce similar small displacements. Positions relative to the footprints of buildings were used to avoid the former effect, and care was taken to centralize the point of interest in the screen, while placing markers, to minimize the latter. Since Geogebra is free, open source software, the reader can easily confirm this work and try other variations. We would like to thank the authors of Geogebra for the wonderful tool they have made freely available to the public.
The construction we used produces two arcs of equal radius separated by a stated interval along a mutual tangent (the roll interval), with one arc tangent to the path of approach and the other passing through a way point and the target. Deriving the construction is left as an interesting exercise for the reader.