7. Snap, angular

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 = 12° (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 4.68 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 (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	Q	v_snap
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

R_bullet	v_snap	R_head	v_snap
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

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 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	1.14	0.29	Small
m_bullet	1.00	0.29	Small
R_bullet	1.00	1.63	Large
Negative effect on snap
Q	-0.05	0.20	Small
v_bulletafter	-0.13	1.20	Medium
R_head	-1.00	2.19	Large
m_head	-1.01	2.84	Large