How to Calculate the Angular Velocities of a Target

[Joe Elliott]

How to Calculate the Angular Velocities of a Target as Seen from any Hypothetical Sniper Position.

About a year and a half ago, someone on this board told me about an article by Dr. Nicholas R. Nalli on the “Sniper Target Tracking Analysis of John F. Kennedy”. I don’t know who this person is because when this website went down, all my messages went away, and I don’t remember his name. He also said that the estimates that Dr. Nalli made seemed to be off, which I agree, they are off. I want to thank that person for sending that message and would like to know who it was.

Dr. Nalli’s article can be seen at:

https://www.acsr.org/wp-content/uploads/2018/12/2018-Target-Tracking-Analysis-of-JFK-Assassination-Nalli.pdf


I have worked on the math on how to calculate the angular velocity of a moving target, like the limousine, from any hypothetical sniper position.


I have had some free time lately so I organized my work on the mathematics on this as follows. Below is a summary of my conclusions:


For five hypothetical shots, the overall angular velocity in degrees per second is:

                       angular
                       velocity
                       my
                       estimate
Position     
TSBD-SN        z-153   4.8
TSBD-SN        z-222   1.9
TSBD-SN        z-312   0.58
Grassy Knoll   z-312   5.1


I used a method for estimating the angular velocity, that first made the following 5 estimates:

1.   Speed of the limousine, called ‘V’.
2.   Distance between the shooting position and the limousine, through 3-D space, called ‘R’.
3.   The difference in the horizontal angle between the direction of the limousine, as seen from the shooting position, and the horizontal direction of travel of the limousine. I name this angle after the Greek letter ‘theta’.
4.   The vertical angle, or the ‘elevation’, of the target as seen from the shooting position. I name this angle after the Greek letter ‘delta’.
5.   The vertical angle, or the ‘vertical bearing’, of the direction of travel of the limousine. I name this angle after the Greek letter ‘gamma’. ‘gamma’ is always -3 degrees because that is roughly the slope of Elm Street.

I use Greek letters to represent various angles:

‘alpha’ is used to specify the azimuth, the horizontal direction of the limousine as seen from the shooting position.

‘beta’  is the ‘horizontal bearing’, the direction of travel of the target in the horizontal plane.

‘theta’ is the difference between α and β.

‘delta’ is the ‘elevation’, the direction of the target in the vertical plane, as seen from the position of the shooter.

‘gamma’ is the ‘vertical bearing’, the direction of travel of the target in the vertical plane.

‘omega’ is the overall difference in direction, in 3-D space, of the target as seen from the sniper’s nest, at two different times.
Like the difference in direction between time “3 seconds” and time “4 seconds” might be 3 degrees, which would indicate an average overall angular velocity of 3 degrees per second during that interval.


‘alpha’ and ‘delta’ are the current horizontal and vertical direction of the target, as seen from the shooter’s position.

‘beta’ and ‘gamma’ are the current horizontal and vertical direction of travel of the limousine. They do not affected by the location of the shooter. A ‘beta’ compass direction of 240 degrees is not affected by the location of the shooter.


‘alpha’, beta’, theta’ are horizontal angles.
‘delta’ and ‘gamma’ are vertical angles.
‘omega’ is neither a horizonal nor a vertical angle but an overall angle in 3-D space.


The three angular velocities I want to calculate are:

d alpha / dt   -   is the horizontal component of the angular velocity.

d delta / dt   -   is the vertical component of the angular velocity.

d omega /dt   -   shows the overall angular velocity of the limousine.


The three equations I use are:

1.   The horizontal component of the angular velocity of the target:

d alpha /dt = (V/R) cos(gamma) sin(theta) / cos(delta)


2.   The vertical component of the angular velocity of the target:

d delta /dt = (V/R) [ cos(delta) sin(gamma) - sin(delta) cos(gamma) cos(theta) ]


3.   The overall angular velocity of the target:

d omega /dt = (V/R) sin (2   asin [
    sqrt ( 2 [ + 1 - cos(theta) cos(delta) cos(gamma) - sin(delta) sin(gamma) ] ) / 2 ]   )


These three equations look a little awkward because I had to spell out the Greek letters, because I could not post text containing Greek letters.


My summary with charts and diagrams is shown on the following PDF file I have online:

https://ibb.co/jR68G6j


The way these three equations are calculated is shown in the following three PDF files I have online:

https://ibb.co/jrVWJv6
https://ibb.co/RPdhTj5
https://ibb.co/bB6MqJ8

And the maps showing the horizontal angles are shown below for shots fired from the Texas School Book Depository and the Grassy Knoll are shown in these 5 PDF files I have online:

For Zapruder Frame 153, for a hypothetical shot from the TSBD sniper’s nest:
https://ibb.co/fSmNfz5

For Zapruder Frame 222, for a hypothetical shot from the TSBD sniper’s nest:
https://ibb.co/PjtL5Hm

For Zapruder Frame 312, for a hypothetical shot from the TSBD sniper’s nest:
https://ibb.co/bJMFsBZ

For Zapruder Frame 312, for a hypothetical shot from the Grassy Knoll, “Gunsmoke” position:
https://ibb.co/LgYf8Bs

For Zapruder Frame 312, for a hypothetical shot from the Grassy Knoll, “Badgeman” position:
https://ibb.co/FHnyymC


It is a little daunting to check through 40 some pages to confirm these three equations are valid. An easier way is to make an Excel spreadsheet that in 3-D space has the location of a sniper in (x,y,z) coordinates. Then select the ‘V’, ‘R’, ‘theta’, ‘delta’ and ‘gamma’ for this example. Use this to calculate the target’s location. And to calculate the target’s location one millisecond later. Then use trigonometry to calculate the change in the azimuth, elevation, and the angle between these two points. Multiple each by 1,000 to see in one gets a close approximation of what one gets with the three equations. Do this 1000 times, which is easy with Excel, and one should get excellent confidence in these equations.

Using the two methods on 1,000 sets of random numbers generated by Excel:
     ‘V’ or velocity (range from 0 to 25 feet per second)
     ‘R’ or range    (range from 100 to 300 feet)
     ‘theta’         (range from -180 through 180 degrees)
     ‘delta’         (range from -30 through 0 degrees)
     ‘gamma’         (range from -10 through 0 degrees)

1.   Using my three equations to calculate:
d theta / dt
d delta / dt
d gamma / dt

2.   Calculating the change in ‘theta’, ‘delta’ and ‘gamma’ over 1 millisecond of time and multiplying by 1000.

I found that the two methods gave basically the same result. The largest difference in any of the three rates over 1000 sets of random numbers was 0.00165 degrees per second, which gives me great confidence that the three equations are accurate.

[John Mytton]

Thanks Joe, here's a little graphic showing how hard it would be tracking a shot down Houston(1+2), then on to a virtually impossible shot when the Limo was whizzing by the front of the building(3) and finally onto the easiest shot when the Limo was travelling down Elm Street(4).



JohnM

[Joe Elliott]

Thanks Joe, here's a little graphic showing how hard it would be tracking a shot down Houston(1+2), then on to a virtually impossible shot when the Limo was whizzing by the front of the building(3) and finally onto the easiest shot when the Limo was travelling down Elm Street(4).



JohnM

Hello John

This is the best graphic of this nature I ever saw. If a picture, or a film is worth a thousand words, or a hundred equations, yes, I admit these graphics are much easier to understand then my equations.


I think it would help if the four different views came with more information, list the Zapruder frame number for the appropriate example, like Example 4.

Example 1 shows two shots, one when the limousine first turns onto Houston and another on Houston. How far is the limousine from the sniper’s nest for the ‘Second’ shot of Example 1?

Example 2: How close is the shot in this example? I would guess about 120 feet on a 2-D map.

Example 3: Is this the ‘Max Holland’ shot?

Example 4: I would guess this is the Zapruder frame 313 shot.


I think it would help if the graphics had text information with them, that would give this sort of information.

Examples for Zapruder frames 153 and 222 would be good additions.

So much for the CT claim that a shot down Houston Street would be so easy because the target is moving right toward him.

Joe

[Brian Roselle]

Joe,

I think I was the one who contacted you. I wanted to compare some numbers to what you did.

I was looking at some possible reasons a first shot might have missed the whole limo, and hypothesized that if Oswald wasn’t a great shot, he still was probably a good enough shot to get close to the intended target, and if he missed the whole limo, it was probably not a real bad miss but likely a minimum limo miss and likely related to the target’s angular velocity.

Back then I had some suspicions the first shot miss was earlier than z150 and possibly as early as near z133. So I used that area as a first shot miss point, looked at estimating angular velocities.
I recall that the estimates were close to yours using the method I used, although the first shot number was naturally a little different. Here were the numbers.
 
Z-frame   Target Angular Velocity Estimated at Z-frame (deg/sec)    Inches the shot missed From Center of Target
Z313                                     0.61                                                               ~ 1-2
Z223                                     1.68                                                               ~ 8
First shot                              5.4                                                               ~ 36


https://drive.google.com/file/d/1Lfn50uzSQmGLF5o2BexNamNsQRJ1SpFY/view?usp=sharing

What was interesting in those numbers was that in this scenario, the amount Oswald missed, per rate of target angular velocity, is similar for all three shots (all fall nearly on a straight line when plotted as seen in the link). That would mean if he had a constant penchant to miss his target “proportional to the angular velocity of the target”, it would not be that surprising to have a minimal limo miss on an up close shot, as opposed to the nearly universal thinking that there was “no way in hell he could have missed the entire limo on the closest shot”.

The 3 feet (or 36 inches) minimum limo miss came from the estimate in this next link which indicated the aiming error, for this scenario’s first shot miss, only had to be 0.5 inches.
https://drive.google.com/file/d/1hZEgKoRdXBBpzrLArLUZh9JsNA3oIE_E/view?usp=sharing

[Joe Elliott]

Joe,

I think I was the one who contacted you. I wanted to compare some numbers to what you did.

Yes. I recognized your name as soon as I saw it. You sent a message to me about the Dr. Nicholas Nalli article on the Target Analysis. I wouldn’t have heard of it if you didn’t alert me. That was about 18 months ago as I recall. It took a long time for me to develop my equations, but most of the time was spent playing Sudoko. And the time spent on my ‘article’ was largely spent on the diagrams, which I am not good at making.

I was looking at some possible reasons a first shot might have missed the whole limo, and hypothesized that if Oswald wasn’t a great shot, he still was probably a good enough shot to get close to the intended target, and if he missed the whole limo, it was probably not a real bad miss but likely a minimum limo miss and likely related to the target’s angular velocity.

Back then I had some suspicions the first shot miss was earlier than z150 and possibly as early as near z133. So I used that area as a first shot miss point, looked at estimating angular velocities.
I recall that the estimates were close to yours using the method I used, although the first shot number was naturally a little different. Here were the numbers.
 
Z-frame   Target Angular Velocity Estimated at Z-frame (deg/sec)    Inches the shot missed From Center of Target
Z313                                     0.61                                                               ~ 1-2
Z223                                     1.68                                                               ~ 8
First shot                              5.4                                                               ~ 36


https://drive.google.com/file/d/1Lfn50uzSQmGLF5o2BexNamNsQRJ1SpFY/view?usp=sharing

What was interesting in those numbers was that in this scenario, the amount Oswald missed, per rate of target angular velocity, is similar for all three shots (all fall nearly on a straight line when plotted as seen in the link). That would mean if he had a constant penchant to miss his target “proportional to the angular velocity of the target”, it would not be that surprising to have a minimal limo miss on an up close shot, as opposed to the nearly universal thinking that there was “no way in hell he could have missed the entire limo on the closest shot”.

The 3 feet (or 36 inches) minimum limo miss came from the estimate in this next link which indicated the aiming error, for this scenario’s first shot miss, only had to be 0.5 inches.
https://drive.google.com/file/d/1hZEgKoRdXBBpzrLArLUZh9JsNA3oIE_E/view?usp=sharing

Your estimates are close to mine. And I suspect the differences may be mostly, or entirely, due to different estimates for the speed of the limousine and the estimate of the angles, and not so much with the math.

Yes, the amount of the miss happens to be close to a linear relationship. But other formulas are possible. Maybe the average angular error is related to the square of the angular velocity of the target. Or abruptly changes above a certain angular velocity. And the relationship probably varies for different shooters. And the relationship would likely change as a shooter got more practice at shooting at a moving target. It would take a lot of real-world testing to find out.

We are among the relatively few LNers with an interest in this obscure subject. And no CTers because it does not support their Grassy Knoll scenario, which doesn’t make sense, from an ‘Angular Velocity of the Target’ point of view.