This post is the third in a series examining the recently released witness flight path math effort by P4T. The previous two linked below lay the groundwork for this post.
Centripetal Acceleration
Banking Equation
In their latest endeavor, P4T presented a number of flight path options summarized in this chart.
Path #2 best fits the NoC eyewitness accounts, but P4T opted to inject the other three when the ideal path fit came close to violating aerodynamic principles. Further, others examining the radius value for #2 question the value used by P4T. So before continuing the math discussion, it is important to define the path being examined.
Regardless of the attempts by P4T to confuse the issue by injecting the additional 3 paths, there is only one which was the subject of this thread. That is the flight path as described by CIT and P4T for several years now based on the eyewitness accounts of individuals located at the Citgo station. This was covered in great detail by CIT in their video release, "The Pentacon". A
video released by the FAA (although created by NORAD), closely matches this eyewitness path and has been heavily touted by P4T as "proof" of that path.
SGT William Lagasse is the eyewitness who started the NoC speculation with his very early accounts that he saw the starboard side of the plane from his location at the Citgo station. His account weighed heavily in The Pentacon video and as observed on the Citgo video his location under the canopy limited his view. So any NoC path has to also weigh heavily on his and SGT Brooks account. In the image below, I have marked the Lagasse/Brooks observed position and fitted a circle beginning with Paik and ending with the impact area. The resulting curved path is the best fit circular curve that minimizes g-forces and assumes constant speed and bank (the scenario presented by P4T). It must be emphasized that it is this NoC path that has been so heavily promoted by both CIT and P4T that is under discussion here, NOT the alternative paths now promoted by P4T to get out of a mathematical quandary.
There is a Chord represented by the distance from Edward Paik's shop to the impact point at the Pentagon, [latex]$$ c $$[/latex] which is ~3960 feet long. The line representing the greatest perpendicular path from the Chord to the curve is the sagitta, [latex]$$ s $$[/latex] which is ~510 feet. P4T presented an equation briefly for solving for the sagitta of the circle represented by the curve, but it is the radius being sought and I prefer deriving the value using common methods and showing my work.
The radius, [latex]$$ R $$[/latex] of the circle is simply the hypotenuse of a right triangle created with one side represented by [latex]$$ \frac{c}{2} $$[/latex] and the other equal to [latex]$$ R - s $$[/latex]. Using the Pythagorean theorem from basic trigonometry, the value of [latex]$$ R $$[/latex] can be found.
[latex]$$ R^2 = {\left({\frac{c}{2}}\right)}^2 + {\left({R - s}\right)^2 $$[/latex],
[latex]$$ R^2 = \frac{c^2}{4} + {R^2} - {2Rs} + {s^2} $$[/latex],
[latex]$$ 2Rs = \frac{c^2}{4} + {s^2} $$[/latex],
[latex]$$ R = \frac{c^2}{8s} + \frac{s}{2} $$[/latex].
Plugging in the values gives [latex]$$ R = 4099 ft $$[/latex] which by using already derived formulas for centripetal acceleration and bank angle gives the following:
Bank angle = 78 degrees
g-force = 4.6 g's
The equation I derived can be verified using the P4T equation with a handy web calculator found
here. The P4T/CIT NoC eyewitness fight path
fails even the simple mathematical model presented by P4T. I can take the path to the next level, but is it really worth the effort?
Prediction: P4T will now throw Lagasse and Brooks under the bus with Morin and Boger.
