Author's Warning: There have already been two threads on this topic. The first was binned due to abusive posts, and the second is under administrative review. I would very much wish to avoid this, and the odium that comes with it.
In this spirit, the post below deals strictly with measurement and calculation and is of an informational nature ONLY. I will not engage in nor tolerate bickering or Internet rivalry of any kind. If you do not wish to discuss the physics or measurements relative to this problem, please find another thread.
Support Your Local Moderator. Thank you for your consideration.
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1. Motivation
It has recently been suggested that the terminal trajectory of American Flight 77 -- the aircraft that impacted the Pentagon on September 11th -- as reconstructed by (a) nearby obstacles, and (b) fragmentary data from the Flight Data Recorder, is implausible given the performance limits of the aircraft. Some have further suggested that this leads one to conclude that a coverup of one form or another is underway.
In this whitepaper, we will examine the measurements and the mathematics in an effort to verify or refute these claims.
2. Boundary Conditions
The proposed boundary conditions of our problem are not well summarized, but can be extracted from articles and follow-up discussion on the Internet, initiated by a group known as "Pilots for 9/11 Truth." One such posting is found here. From this discussion, we derive the following boundary conditions on our aircraft:
3. Model Trajectory
At this time we do not have any insight into the aircraft altitude, rate of descent, or attitude apart from the conditions listed above. To provide a simple trajectory model, we use the following assumptions:
where y is the altitude, x is the distance downrange from the Pentagon, s is the slope of the parabola, v is the horizontal position of the vertex, and h is the height of the vertex. All variables are in units of feet except for s, which is in units of feet-1.
In order to estimate the aircraft performance requirements, we must solve this equation for each of the various cases, which are listed below in tabular form. The vertex horizontal position is left as v because it is a variable that must be solved. Point 1 is the contact point with the light pole, Point 2 is the position directly above the radio tower. All vertical dimensions are given in feet ASL, referenced to the bottom of the aircraft.
Case
||
Vertex
||
Point 1
||
Point 2
|| || ||
A ||{ v , 33}||{1016, 80}||{3400, 304}
B ||{ v , 39}||{1016, 80}||{3400, 304}
C ||{ v , 39}||{1016, 59.4}||{3400, 304}
D ||{ v , 39}||{1016, 59.4}||{3400, 408}
E ||{ v , 39}||{1016, 59.4}||{3400, 480}
F ||{ v , 39}||{1016, 59.4}||{3400, 699}
4. Solution
Solution of the problem is straightforward -- by substituting the vertex height into the equation above, then substituting both Point 1 and Point 2 into the equation, one can extract both the horizontal location of the vertex v and the slope s from the results.
What is ultimately desired, however, is a measure of how much stress was felt by the airframe. This can be computed from s, as follows:
The true vertical acceleration of the aircraft is equal to the second time derivative of the altitude y. However, we need to re-express the equation in terms of time. Since we have assumed a constant horizontal velocity, we can re-express x in terms of time:
where t is the time, in seconds, after the aircraft passes the radio tower. We can then substitute this into equation (1) to find the altitude as a function of time y(t).
Once this is done, the vertical acceleration "y double dot" is equal to the second derivative, which is the following:
Therefore, once we know s, we know the acceleration of the aircraft.
The following figures, drawn to scale, summarize the input conditions, and overlay this with the solution of the most difficult case (Case F) respectively. Click on the images to view them in larger format.
The figures demonstrate that the minimum effort trajectory -- to just barely clear the radio tower, clip the pole, and strike the Pentagon -- is actually quite shallow. Likewise, even the most severe trajectory (plotted in the lower figure) does not appear unreasonable. This is confirmed by the numerical results, which we now present. The solution to the various cases is as follows, with all measurements rounded to three significant digits:
Case
||
Vertex v
||
s (1/feet)
||
Accel (ft/sec2)
|| || ||
A ||-685 ft||1.62x10 -5 ||19.8
B ||-528 ft||1.72x10 -5 ||21.0
C ||103 ft||2.45x10 -5 ||29.9
D ||284 ft||3.81x10 -5 ||46.5
E ||363 ft||4.78x10 -5 ||58.3
F ||507 ft||7.87x10 -5 ||96.0
5. Discussion
First, let us consider the input. The various cases simulated here are of varying credibility with respect to the input conditions.
Next, let us consider the results of the various cases:
6. Conclusions
All of the available data suggests a terminal trajectory that is achievable by a Boeing 757 aircraft. Even the most unfavorable example suggested by "Pilots for 9/11 Truth," specifying an initial height inconsistent with the FDR figures supplied by them along with the most challenging altitude at both light pole and impact, requires only 4.0 g of load in the airframe for a mere 4.4 seconds. The aircraft is expected to survive such a load without any significant risk of failure.
Based on these calculations, there is absolutely no case to be made that (1) the obstacles are inconsistent with the impact of Flight 77, (2) the FDR data is inconsistent with the impact of Flight 77, or (3) the FDR data is inconsistent with impacts to the obstacles themselves. Furthermore, with the exception of Case F, all of the various requirements lead to a trajectory that is easily reconcilable with an amateur pilot at the controls. Even Case F is plausible, it is merely unexpected.
Disclaimer
The preceding is my opinion alone and does not represent the position of any agency, public or private. All work done with my own materials on my own time. This message contains no schematics, technology, or recommended flight paths of any sort. Always follow all FAA regulations and guidelines when operating any aircraft, and avoid contact with any and all ground obstacles at all times.
In this spirit, the post below deals strictly with measurement and calculation and is of an informational nature ONLY. I will not engage in nor tolerate bickering or Internet rivalry of any kind. If you do not wish to discuss the physics or measurements relative to this problem, please find another thread.
Support Your Local Moderator. Thank you for your consideration.
----
1. Motivation
It has recently been suggested that the terminal trajectory of American Flight 77 -- the aircraft that impacted the Pentagon on September 11th -- as reconstructed by (a) nearby obstacles, and (b) fragmentary data from the Flight Data Recorder, is implausible given the performance limits of the aircraft. Some have further suggested that this leads one to conclude that a coverup of one form or another is underway.
In this whitepaper, we will examine the measurements and the mathematics in an effort to verify or refute these claims.
2. Boundary Conditions
The proposed boundary conditions of our problem are not well summarized, but can be extracted from articles and follow-up discussion on the Internet, initiated by a group known as "Pilots for 9/11 Truth." One such posting is found here. From this discussion, we derive the following boundary conditions on our aircraft:
- The impact site at the Pentagon was no lower than 33 feet above mean sea level (33 feet ASL). We further insist that the aircraft (referenced to the bottom of the fuselage) cannot pass below this line at any time.
- There was a light pole, struck by the aircraft, placed approximately 1,016 feet away from the Pentagon along the line of travel. This light pole reached a height of 80 feet ASL.
- There was also a substantial radio tower, operated by the Virginia Highway Patrol, reaching a total height of 304 feet ASL further back along the line of travel. This tower was located roughly 3,400 feet downrange. The aircraft did not destroy this tower, though accounts vary as to whether it missed the tower entirely or brushed it slightly.
- The aircraft's last known velocity was approximately 781 feet per second. We will assume this is the groundspeed in all cases. For shallow angles of pitch, this is approximately constant; we further have no insight into thrust or drag in the final few seconds before impact.
- Assuming the impact was exactly at ground level seems too strict. For Case B and following, assume the aircraft retained six feet of margin, thus contacting the Pentagon at 39 feet ASL rather than 33 feet, again referenced to the bottom of the aircraft. This defines the new lowest point of the trajectory. The choice of six feet is arbitrary but is equal to about half the height of one story of the Pentagon, and is half the diameter of the fuselage.
- It has been suggested that the light pole was not struck at the very top, but instead hit in the middle, at roughly 59.4 feet ASL. Use this assumption for Case C and all subsequent cases.
- FDR data, which stopped at approximately the distance of the radio tower, suggests the actual altitude at that moment was 408 feet (using RADALT data) or 480 feet (possibly using a moving average or air data). These altitudes at the radio tower, 408 feet ASL and 480 feet ASL, are Case D and Case E respectively.
- Finally, the NTSB animation -- which does not appear to be calibrated for this purpose -- suggests a height of 699 feet ASL as it passed over the tower. This is Case F.
3. Model Trajectory
At this time we do not have any insight into the aircraft altitude, rate of descent, or attitude apart from the conditions listed above. To provide a simple trajectory model, we use the following assumptions:
- The aircraft rate-of-descent at the radio tower is unrestricted. Only altitudes are prescribed, along with a fixed horizontal velocity of 781 feet per second.
- We assume the aircraft exerts a constant pull-up maneuver, beginning at the radio tower, and ending when the rate of descent reaches zero.
- This point is the vertex of a parabolic curve. If the aircraft reaches the vertex prior to striking the Pentagon, it continues flat and level from the vertex until impact.
(1): y = s(x - v)2 + h
where y is the altitude, x is the distance downrange from the Pentagon, s is the slope of the parabola, v is the horizontal position of the vertex, and h is the height of the vertex. All variables are in units of feet except for s, which is in units of feet-1.
In order to estimate the aircraft performance requirements, we must solve this equation for each of the various cases, which are listed below in tabular form. The vertex horizontal position is left as v because it is a variable that must be solved. Point 1 is the contact point with the light pole, Point 2 is the position directly above the radio tower. All vertical dimensions are given in feet ASL, referenced to the bottom of the aircraft.
|| || ||
A ||{ v , 33}||{1016, 80}||{3400, 304}
B ||{ v , 39}||{1016, 80}||{3400, 304}
C ||{ v , 39}||{1016, 59.4}||{3400, 304}
D ||{ v , 39}||{1016, 59.4}||{3400, 408}
E ||{ v , 39}||{1016, 59.4}||{3400, 480}
F ||{ v , 39}||{1016, 59.4}||{3400, 699}
4. Solution
Solution of the problem is straightforward -- by substituting the vertex height into the equation above, then substituting both Point 1 and Point 2 into the equation, one can extract both the horizontal location of the vertex v and the slope s from the results.
What is ultimately desired, however, is a measure of how much stress was felt by the airframe. This can be computed from s, as follows:
The true vertical acceleration of the aircraft is equal to the second time derivative of the altitude y. However, we need to re-express the equation in terms of time. Since we have assumed a constant horizontal velocity, we can re-express x in terms of time:
(2): x = x0 - 781 t feet / second
where t is the time, in seconds, after the aircraft passes the radio tower. We can then substitute this into equation (1) to find the altitude as a function of time y(t).
Once this is done, the vertical acceleration "y double dot" is equal to the second derivative, which is the following:
(3): "y double dot" = d2/dt2 y(t) = 2 s (781 feet/second)2
Therefore, once we know s, we know the acceleration of the aircraft.
The following figures, drawn to scale, summarize the input conditions, and overlay this with the solution of the most difficult case (Case F) respectively. Click on the images to view them in larger format.
The figures demonstrate that the minimum effort trajectory -- to just barely clear the radio tower, clip the pole, and strike the Pentagon -- is actually quite shallow. Likewise, even the most severe trajectory (plotted in the lower figure) does not appear unreasonable. This is confirmed by the numerical results, which we now present. The solution to the various cases is as follows, with all measurements rounded to three significant digits:
|| || ||
A ||-685 ft||1.62x10 -5 ||19.8
B ||-528 ft||1.72x10 -5 ||21.0
C ||103 ft||2.45x10 -5 ||29.9
D ||284 ft||3.81x10 -5 ||46.5
E ||363 ft||4.78x10 -5 ||58.3
F ||507 ft||7.87x10 -5 ||96.0
5. Discussion
First, let us consider the input. The various cases simulated here are of varying credibility with respect to the input conditions.
Impact Height
It is not known precisely where the aircraft hit the Pentagon, and the ground is not perfectly flat in any event. Case B is therefore considered somewhat more reasonable than Case A. However, if the model predicts a vertex behind the Pentagon, i.e. the aircraft is still descending upon impact, then either situation may be plausible.
Others will also note that the aircraft engines droop well below the fuselage, and thus an even higher terminal altitude would seem preferable -- this is not strictly correct. It should be kept in mind that while flying (and particularly during a hard pull-up maneuver) the wings would be flexed upwards, and engines along with them, thus it is not clear that the nacelles would in fact be below the fuselage. Finally, we have not rigorously defined "the bottom of the aircraft," nor can we say with any confidence whether fuselage, engines, wings, or all three are candidates to strike the light pole. At the resolution of this analysis, and given other uncertainties in the measurements, this effect is thus disregarded and should simply be carried as an uncertainty of a few feet.
Light Pole Contact
We are all agreed that the aircraft did contact the light pole, but it is not certain exactly where the pole was struck. The argument for a lower contact is based on crumpling or buckling of the pole, which left it bent somewhere in the middle. It is possible for this to have been caused by direct impact, but also by buckling away from the contact point or secondary impact -- thus either Case B or Case C seems equally acceptable.
Radio Tower Contact
As mentioned above, there is circumstantial evidence that the radio tower was, in fact, lightly struck by the aircraft. We cannot be certain of this one way or the other. Case C and Case D (and by inference, Case B) should be treated as equally acceptable.
Variation in FDR Instruments
The two FDR measurements disagree by about 70 feet of altitude. This is not at all surprising given the speed and maneuver in play -- even a single second of latency between the measurements could easily account for this discrepancy. Case E, however, makes contact with the radio tower impossible, and the evidence biases us slightly towards the opinion that there was contact or that it was very close. Therefore, Case D is preferred to Case E, but either one should be treated as credible.
NTSB Animation
As stated previously, there is no evidence that the NTSB animation was intended for use in precise trajectory estimation. We do not know how it was calibrated or whether it was calibrated at all. Case F is also an outlier, compared to the FDR data as well as the possible contact with the radio tower. Case F is therefore viewed as credible but unlikely.
It is not known precisely where the aircraft hit the Pentagon, and the ground is not perfectly flat in any event. Case B is therefore considered somewhat more reasonable than Case A. However, if the model predicts a vertex behind the Pentagon, i.e. the aircraft is still descending upon impact, then either situation may be plausible.
Others will also note that the aircraft engines droop well below the fuselage, and thus an even higher terminal altitude would seem preferable -- this is not strictly correct. It should be kept in mind that while flying (and particularly during a hard pull-up maneuver) the wings would be flexed upwards, and engines along with them, thus it is not clear that the nacelles would in fact be below the fuselage. Finally, we have not rigorously defined "the bottom of the aircraft," nor can we say with any confidence whether fuselage, engines, wings, or all three are candidates to strike the light pole. At the resolution of this analysis, and given other uncertainties in the measurements, this effect is thus disregarded and should simply be carried as an uncertainty of a few feet.
Light Pole Contact
We are all agreed that the aircraft did contact the light pole, but it is not certain exactly where the pole was struck. The argument for a lower contact is based on crumpling or buckling of the pole, which left it bent somewhere in the middle. It is possible for this to have been caused by direct impact, but also by buckling away from the contact point or secondary impact -- thus either Case B or Case C seems equally acceptable.
Radio Tower Contact
As mentioned above, there is circumstantial evidence that the radio tower was, in fact, lightly struck by the aircraft. We cannot be certain of this one way or the other. Case C and Case D (and by inference, Case B) should be treated as equally acceptable.
Variation in FDR Instruments
The two FDR measurements disagree by about 70 feet of altitude. This is not at all surprising given the speed and maneuver in play -- even a single second of latency between the measurements could easily account for this discrepancy. Case E, however, makes contact with the radio tower impossible, and the evidence biases us slightly towards the opinion that there was contact or that it was very close. Therefore, Case D is preferred to Case E, but either one should be treated as credible.
NTSB Animation
As stated previously, there is no evidence that the NTSB animation was intended for use in precise trajectory estimation. We do not know how it was calibrated or whether it was calibrated at all. Case F is also an outlier, compared to the FDR data as well as the possible contact with the radio tower. Case F is therefore viewed as credible but unlikely.
Next, let us consider the results of the various cases:
Case A and Case B
Forcing the aircraft to pull up six feet higher than minimum causes the aircraft to flatten out before impact. In both cases, the aircraft is still descending when it contacts the Pentagon -- a situation consistent with an amateur crash approach. Case A experiences only 0.62 g of acceleration, or a total of 1.62 g of g-loading; Case B implies 0.66 g and 1.66 g respectively.
Both of these cases are not only within the Boeing 757-233's performance envelope, but in fact not terribly unusual in ordinary operation, apart from the altitude. Such a low stress level would not imperil any aircraft.
Case B and C
Retaining the higher impact point and forcing the plane to hit the light pole at a lower altitude forces the aircraft to dive more initially, then pull up shorter and sharper. The acceleration increases from 0.66 to 0.93 g (1.93 g airframe stress). This figure is still easily within the performance envelope of the aircraft, and not difficult for an amateur pilot.
Case C and D
Raising the trajectory over the radio tower implies a higher initial descent, but this cannot be countered quickly since we still require contact with the light pole at its midpoint. Acceleration for Case D increases to 1.45 g (2.45 g load on the airframe), unusual in normal operation but still well within the aircraft's capabilities.
The aircraft still pulls up a mere 284 feet (about a third of a second) before impact, which implies a similar aim point and thus is still entirely credible for a suicide pilot. Case C and D thus appear equally credible on all counts.
Case D and E
The slightly higher altitude over the radio tower has only a slight effect, with Case E requiring 1.82 g of vertical acceleration (2.82 g of load). The aircraft also pulls up a mere 80 feet sooner. This figure is still within the performance limits. Cases D and E are basically indistinguishable.
Case F
Our most extreme estimate, raising the trajectory by yet another 220 feet over the radio tower, gives us a result of 3.0 g vertical acceleration, or 4.0 g of load. This is greater than the design specification of the aircraft, however this is below the ultimate strength of the aircraft, and there are numerous examples of passenger jets that have experienced even higher loads in flight without being destroyed. Because the aircraft must only experience this load for a few seconds, it should be viewed as entirely possible.
This trajectory also calls for the aircraft to pull up over 500 feet away from impact. This should not be necessary. Therefore, If this trajectory is accurate, it suggests the terrorists dived the aircraft much too steeply, only recovering in the last few seconds. Given trials of inexperienced pilots in simulators recreating this event, we expect a flatter terminal approach -- the light pole probably would not have been struck had they started so high. Also, we must take into account the imprecise nature of the NSTB simulation on which this case is founded. We therefore consider Case F to be unlikely. Nonetheless, even this case is definitely within the capability of the aircraft.
Forcing the aircraft to pull up six feet higher than minimum causes the aircraft to flatten out before impact. In both cases, the aircraft is still descending when it contacts the Pentagon -- a situation consistent with an amateur crash approach. Case A experiences only 0.62 g of acceleration, or a total of 1.62 g of g-loading; Case B implies 0.66 g and 1.66 g respectively.
Both of these cases are not only within the Boeing 757-233's performance envelope, but in fact not terribly unusual in ordinary operation, apart from the altitude. Such a low stress level would not imperil any aircraft.
Case B and C
Retaining the higher impact point and forcing the plane to hit the light pole at a lower altitude forces the aircraft to dive more initially, then pull up shorter and sharper. The acceleration increases from 0.66 to 0.93 g (1.93 g airframe stress). This figure is still easily within the performance envelope of the aircraft, and not difficult for an amateur pilot.
Case C and D
Raising the trajectory over the radio tower implies a higher initial descent, but this cannot be countered quickly since we still require contact with the light pole at its midpoint. Acceleration for Case D increases to 1.45 g (2.45 g load on the airframe), unusual in normal operation but still well within the aircraft's capabilities.
The aircraft still pulls up a mere 284 feet (about a third of a second) before impact, which implies a similar aim point and thus is still entirely credible for a suicide pilot. Case C and D thus appear equally credible on all counts.
Case D and E
The slightly higher altitude over the radio tower has only a slight effect, with Case E requiring 1.82 g of vertical acceleration (2.82 g of load). The aircraft also pulls up a mere 80 feet sooner. This figure is still within the performance limits. Cases D and E are basically indistinguishable.
Case F
Our most extreme estimate, raising the trajectory by yet another 220 feet over the radio tower, gives us a result of 3.0 g vertical acceleration, or 4.0 g of load. This is greater than the design specification of the aircraft, however this is below the ultimate strength of the aircraft, and there are numerous examples of passenger jets that have experienced even higher loads in flight without being destroyed. Because the aircraft must only experience this load for a few seconds, it should be viewed as entirely possible.
This trajectory also calls for the aircraft to pull up over 500 feet away from impact. This should not be necessary. Therefore, If this trajectory is accurate, it suggests the terrorists dived the aircraft much too steeply, only recovering in the last few seconds. Given trials of inexperienced pilots in simulators recreating this event, we expect a flatter terminal approach -- the light pole probably would not have been struck had they started so high. Also, we must take into account the imprecise nature of the NSTB simulation on which this case is founded. We therefore consider Case F to be unlikely. Nonetheless, even this case is definitely within the capability of the aircraft.
6. Conclusions
All of the available data suggests a terminal trajectory that is achievable by a Boeing 757 aircraft. Even the most unfavorable example suggested by "Pilots for 9/11 Truth," specifying an initial height inconsistent with the FDR figures supplied by them along with the most challenging altitude at both light pole and impact, requires only 4.0 g of load in the airframe for a mere 4.4 seconds. The aircraft is expected to survive such a load without any significant risk of failure.
Based on these calculations, there is absolutely no case to be made that (1) the obstacles are inconsistent with the impact of Flight 77, (2) the FDR data is inconsistent with the impact of Flight 77, or (3) the FDR data is inconsistent with impacts to the obstacles themselves. Furthermore, with the exception of Case F, all of the various requirements lead to a trajectory that is easily reconcilable with an amateur pilot at the controls. Even Case F is plausible, it is merely unexpected.
Disclaimer
The preceding is my opinion alone and does not represent the position of any agency, public or private. All work done with my own materials on my own time. This message contains no schematics, technology, or recommended flight paths of any sort. Always follow all FAA regulations and guidelines when operating any aircraft, and avoid contact with any and all ground obstacles at all times.
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