7. The forward snapangular calculations

    Because JFK's head rotated about the top of his neck after it was hit, it would seem more appropriate to use angular calculations to calculate the forward speed of the snap. The equation is as follows:

 

where mbullet, mhead, vbullet, Q, and vbulletafter are the same as before, Rhead is the radius of the head, and Rbullet is the lever arm of the bullet about the point of rotation, here assumed to be the top of the neck. Starting values for the variables are mbullet = 161 grains, vbullet = 1800 ft s-1, mhead = 7 lb, vbulletafter = 200 ft s-1, Q = 12 (these five the same as for the linear calculations), Rbullet = 5.75 in, and Rhead = 4.5 in. The base answer determined with these values is 4.68 ft s-1 for the snap, which is 10% less than from the linear calculations. This value is still consistent with the average speed of 3.3 ft s-1 determined from the Zapruder film, however, for the same three reasons discussed for the linear calculations.

Sensitivity analysis
    The sensitivity analysis is run the same as for the linear case. As before, the standard values are shown in boldface. Also as before, the sensitivities fall into two discrete groups, low values around -0.1 (vbulletafter and Q) and high values of around 1 (mbullet, vbullet, mhead, Rbullet, and Rhead). Again, the most sensitive variable (with the greatest range of effects on the snap), is mhead. This time, however, the second most sensitive variable is Rhead, with a range of effects of 2.1 ft s-1 (as opposed to 2.8 for mhead). It is significant that the two most sensitive variables are properties of the head.
    Thus the angular calculations give results very similar to the linear calculations.

Sensitivity analysis, Snap 1 angular

mbullet

vsnap

vbullet

vsnap

vbulletafter

vsnap

mhead

vsnap

Q

vsnap

156

4.535

1750

4.534

 

 

 

 

7

4.758

157

4.564

1760

4.563

0

5.280

5.0

6.552

8

4.746

158

4.593

1770

4.592

50

5.130

5.5

5.957

9

4.732

159

4.622

1780

4.622

100

4.980

6.0

5.460

10

4.716

160

4.651

1790

4.651

150

4.830

6.5

5.040

11

4.699

161

4.680

1800

4.680

200

4.680

7.0

4.680

12

4.680

162

4.709

1810

4.710

250

4.530

7.5

4.368

13

4.660

163

4.738

1820

4.739

300

4.380

8.0

4.095

14

4.638

164

4.767

1830

4.768

350

4.230

8.5

3.854

15

4.614

165

4.797

1840

4.798

400

4.080

9.0

3.640

16

4.589

166

4.826

1850

4.827

 

 

 

 

17

4.562

Sensitivity = 
[(4.709-4.651)/2]/
[4.680/161] = 1.00

Sensitivity = 
[(4.710-4.651)/20]/
[4.680/1800] = 1.14

Sensitivity = 
[(4.530-4.830)/100]/
[4.680/200] = -0.13

Sensitivity = 
[(4.368-5.040)/1.0]/
[4.680/7.0] = -1.00

Sensitivity = 
[(4.638-4.716)/4]/
[4.680/12] = -0.05

Range = 0.29

Range = 0.29

Range = 1.20

Range = 2.91

Range = 0.20

Rbullet

vsnap

Rhead

vsnap

4.75

3.866

3.50

6.017

5.00

4.070

3.75

5.616

5.25

4.273

4.00

5.265

5.50

4.477

4.25

4.956

5.75

4.680

4.50

4.680

6.00

4.884

4.75

4.434

6.25

5.087

5.00

4.212

6.50

5.291

5.25

4.012

6.75

5.494

5.50

3.829

Sensitivity = 
[(4.884-4.477)/0.50]/
[4.680/5.75] = 1.00

Sensitivity = 
[(4.434-4.956)/0.50]/
[4.680/4.50] = -1.00

Range = 1.63

Range = 2.19

Ordered summary of sensitivities

Variable Sensitivity of vsnap Range of vsnap, ft s-1 Magnitude
Positive effect on snap
vbullet 1.14 0.29 Small
mbullet 1.00 0.29 Small
Rbullet 1.00 1.63 Large
Negative effect on snap
Q  -0.05 0.20 Small
vbulletafter -0.13 1.20 Medium
Rhead -1.00 2.19 Large
mhead -1.01 2.84 Large

    The sensitivities and ranges of effect for the five variables used previously for the linear calculation are nearly the same here. The new rotational variables, Rbullet and Rhead, have large effects, with Rbullet being positive and Rhead negative. The difference in sign is easy to understand. Since Rbullet represents the lever arm of the impacting bullet, greater values will make the head easier to rotate (positive effect). Since Rhead is related to the moment of inertia of the head, greater values will make the head harder to rotate (negative effect).
    The two most important variables here (as defined by range of snap) are the mass and radius of the head (in that order). Just behind that come the lever arm of the bullet and the exit speed of the bullet. Relative to these variables, the mass, velocity, and angle of inclination of the bullet are much less important.

Is the snap quantitatively consistent with a shot from Oswald's rifle?
    We close this section on the snap by returning to the question that prompted it. The answer is a clear yes, that the calculated speed of the snap (4.55 ft s-1) is completely consistent with the observed average speed of >3.3 ft s-1. This very important result allows us to progress to the next question, whether the initial rearward lurch of JFK's head and body are consistent with the same shot.

Ahead to Plausibility Arguments for Lurch
Back to Snap Linear

Back to Physics of the Head Shot