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.pdfI 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/jR68G6jThe way these three equations are calculated is shown in the following three PDF files I have online:

https://ibb.co/jrVWJv6https://ibb.co/RPdhTj5https://ibb.co/bB6MqJ8And 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/fSmNfz5For Zapruder Frame 222, for a hypothetical shot from the TSBD sniper’s nest:

https://ibb.co/PjtL5HmFor Zapruder Frame 312, for a hypothetical shot from the TSBD sniper’s nest:

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

https://ibb.co/LgYf8BsFor Zapruder Frame 312, for a hypothetical shot from the Grassy Knoll, “Badgeman” position:

https://ibb.co/FHnyymCIt 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.