7--SAng

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 m_bullet, m_head, v_bullet, Q, and v_bulletafter are the same as before, R_head is the radius of the head, and R_bullet 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 m_bullet = 161 grains, v_bullet = 1800 ft s^-1, m_head = 7 lb, v_bulletafter = 200 ft s^-1, Q = 17° (these five the same as for the linear calculations), R_bullet = 5.75 in, and R_head = 4.5 in. The base answer determined with these values is 4.562 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 was 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 (v_bulletafter and Q) and high values of around 1 (m_bullet, v_bullet, m_head, R_bullet, and R_head). Again, the most sensitive variable (with the greatest range of effects on the snap), is m_head. This time, however, the second most sensitive variable is R_head, with a range of effects of 2.1 ft s^-1 (as opposed to 2.8 for m_head). 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.

m_bullet	v_snap	v_bullet	v_snap	v_bulletafter	v_snap	m_head	v_snap	R_bullet	v_snap
156	4.421	1750	4.419
157	4.449	1760	4.448	0	5.162	5	6.387	4.75	3.769
158	4.477	1770	4.476	50	5.012	5.5	5.807	5	3.967
159	4.506	1780	4.505	100	4.862	6	5.323	5.25	4.166
160	4.534	1790	4.534	150	4.712	6.5	4.913	5.5	4.364
161	4.562	1800	4.562	200	4.562	7	4.562	5.75	4.562
162	4.591	1810	4.591	250	4.412	7.5	4.258	6	4.761
163	4.619	1820	4.62	300	4.262	8	3.992	6.25	4.959
164	4.647	1830	4.648	350	4.112	8.5	3.757	6.5	5.157
165	4.676	1840	4.677	400	3.963	9	3.548	6.75	5.356
166	4.704	1850	4.706
Sensitivity = [(4.591-4.534)/2]/ [4.562/161] = 1.01		Sensitivity = [(4.591-4.534)/20]/ [4.562/1800] = 1.12		Sensitivity = [(4.412-4.712)/100]/ [4.562/200] = -0.13		Sensitivity = [(4.258-4.913)/1]/ [4.562/7] = -1.01		Sensitivity = [(4.761-4.364)/0.5]/ [4.562/5.75] = 1.00
Range = 0.28		Range = 0.29		Range = 1.12		Range = 2.84		Range = 1.59

R_head	v_snap	Q	v_snap
		12	4.68
3.5	5.866	13	4.66
3.75	5.475	14	4.638
4	5.133	15	4.614
4.25	4.831	16	4.589
4.5	4.562	17	4.562
4.75	4.322	18	4.534
5	4.106	19	4.504
5.25	3.911	20	4.473
5.5	3.733	21	4.44
		22	4.405
Sensitivity = [(4.322-4.831)/0.5]/ [4.562/4.5] = -1.00		Sensitivity = [(4.534-4.589)/2]/ [4.562/17] = -0.102
Sensitivity = [(4.322-4.831)/0.5]/ [4.562/4.5] = -1.00		Sensitivity = [(4.534-4.589)/2]/ [4.562/17] = -0.102
Range = 2.13		Range = 0.28

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, R_bullet and R_head, have large effects, with R_bullet being positive and R_head negative. The difference in sign is easy to understand. Since R_bullet represents the lever arm of the impacting bullet, greater values will make the head easier to rotate (positive effect). Since R_head 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 (4.5–5.0 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.

Variable	Sensitivity of v_snap	*Range of v_snap, ft s^-1*	Magnitude
Positive effect on snap
v_bullet	1.12	0.29	Small
m_bullet	1.01	0.28	Small
R_bullet	1.00	1.59	Large
Negative effect on snap
Q	-0.10	0.28	Small
v_bulletafter	-0.13	1.12	Medium
R_head	-1.00	2.13	Large
m_head	-1.01	2.84	Large