7. The forward snap—angular 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 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.
Sensitivity analysis, Snap 1 angular
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
= 
Sensitivity
= 
Sensitivity
= 
Sensitivity
= 
Sensitivity
= 

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
= 
Sensitivity
= 

Range = 1.63 
Range = 2.19 
Ordered summary of sensitivities
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 
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.
Ahead to Plausibility Arguments for Lurch
Back to Snap Linear
Back to Physics of the Head Shot