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 (4.5–5 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.