16. Lurch 1 Angular—with bullet, head, and cloud
The model and its justification
Motions are of two basic types—translational
and rotational. Although every beginning physics students finds the
translational version more intuitive than the rotational, we really should be
using the rotational here. That is easy to justify, for the immediate effect of
the head shot was to rotate JFK's head forward about his neck, and the later
effect was to rotate his upper torso rearward about his hips. In other
words, JFK rotated rather than translated. We began with the translational
version simply because it is easier to understand. In truth, it should not be
used to address his motions after the head shot.
The basic rotational parameters are analogous to the
translational parameters, as shown in the table below:
Parameter  Translational  Rotational 
Momentum  p = mv  Ω = mvL/3 
Kinetic energy  KE = 0.5mv^{2}  KE = 0.5mv^{2}f_{I}/3 
where L = lever arm (length of the rotating body in this case), v
is the speed at the outer end of the rotating object (the top of Kennedy's head,
for example), and f_{I} is a geometrical factor having to do with
the shape of the object that is rotating (f_{I} is 1.11 for a
rotating rod).
Lurch 1 Angular is the rotational analog of Lurch 1 Linear.
It is the rotational procedure for
calculating the speed of the rearward lurch of upper body from only the bullet, the body, and
the diffuse
cloud of fragments. You just write the rotational versions of the equations of
conservation, which are very similar to the translational versions, and solve
them in the same way. Recall that each term in the conservation equations
below has an L in the numerator, but that it has been eliminated because they
all cancel.
Solving the simultaneous equations
The rotational equations of conservation are shown
below. Note how similar they are to the translational versions in Lurch 1
Linear. The answer for v_{bodyafter} comes out 35%
greater in magnitude than in the linear case —
2.9 ft s^{1}
vs. 2.2 ft s^{1}. Recall that the actual initial lurch was 0.5–1.0
ft s^{1} rearward This confirms that these simple translational and
rotational equations are off.
Default values of variables
m_{bullet} = 161 gr  v_{bullet} = 1800 ft s^{1} 
m_{body} = 85 lb  v_{bulletafter} = 200 ft s^{1} 
m_{cloud} = 0.3 lb  PE = 300 ftlb 
Q = 12°  f_{I} = 1.11 
v_{bodyafter} = 2.92 ft s^{1} v_{cloud} = 425 ft s^{1}
Momentum  Energy, ftlb  
Before  After  Before  After 
Ω_{bullet} = 3.80  Ω_{bulletafter} = 0.43  KE_{bullet} = 1164  KE_{bulletafter} = 14 
Ω_{bodyafter} = 8.58  KE_{bodyafter} = 4  
Ω_{cloud} = 11.95  KE_{cloud} = 846  
PE = 300 
Sensitivity analysis
The sensitivity analysis for Lurch 1 Angular is very
similar to that of Lurch 1 Linear. The most important variable is m_{cloud},
followed at a distance by m_{body} and PE. The other five variables are much
less significant, and don't have to be known particularly well.
Sensitivity tests, Lurch 1 angular
(Standard conditions in boldface)
m_{bullet} 
v_{lurch} 
m_{body} 
v_{lurch} 
m_{cloud} 
v_{lurch} 
v_{bullet} 
v_{lurch} 
v_{bulletafter} 
v_{lurch} 
156 
2.870 




1750 
2.801 


157 
2.880 
65 
3.820 


1760 
2.825 
0 
2.809 
158 
2.891 
70 
3.547 


1770 
2.849 
50 
2.844 
159 
2.901 
75 
3.310 
0.1 
1.204 
1780 
2.873 
100 
2.874 
160 
2.911 
80 
3.104 
0.2 
2.176 
1790 
2.897 
150 
2.900 
161 
2.921 
85 
2.921 
0.3 
2.921 
1800 
2.921 
200 
2.921 
162 
2.931 
90 
2.759 
0.4 
3.549 
1810 
2.945 
250 
2.938 
163 
2.941 
95 
2.613 
0.5 
4.101 
1820 
2.968 
300 
2.951 
164 
2.951 
100 
2.483 
0.6 
4.601 
1830 
2.992 
350 
2.959 
165 
2.960 
105 
2.365 
0.7 
5.059 
1840 
3.015 
400 
2.962 
166 
2.970 


0.8 
5.486 
1850 
3.039 


Sensitivity = 
Sensitivity = 
Sensitivity = 
Sensitivity = 
Sensitivity = 

Range = 0.10 
Range = 1.46 
Range = 4.28 
Range = 0.24 
Range = 0.15 
PE 
v_{lurch} 
Q 
v_{lurch} 
f_{I} 
v_{lurch} 
0 
3.583 




50 
3.479 
7 
2.902 
1.06 
3.058 
100 
3.373 
8 
2.905 
1.07 
3.030 
150 
3.265 
9 
2.908 
1.08 
3.002 
200 
3.153 
10 
2.912 
1.09 
2.974 
250 
3.039 
11 
2.916 
1.10 
2.947 
300 
2.921 
12 
2.921 
1.11 
2.921 
350 
2.800 
13 
2.926 
1.12 
2.895 
400 
2.675 
14 
2.931 
1.13 
2.869 
450 
2.545 
15 
2.937 
1.14 
2.844 
500 
2.411 
16 
2.943 
1.15 
2.820 
550 
2.272 
17 
2.950 
1.16 
2.795 
600 
2.127 




Sensitivity = 
Sensitivity = 
Sensitivity = 

Range = 1.46 
Range = 0.05 
Range = 0.26 
Ordered summary of sensitivities
Variable  Sensitivity of v_{lurch}  Range of v_{lurch}, ft s^{1}  Magnitude 
Positive effect on lurch (reduces rearward velocity)  
f_{I}  0.99  0.26  Small 
PE  0.24  1.46  Large 
m_{body}  1.00  1.46  Large 
Negative effect on lurch (increases rearward velocity)  
Q  0.02  0.05  Small 
m_{bullet}  0.55  0.10  Small 
v_{bulletafter}  0.03  0.15  Small 
v_{bullet}  1.48  0.24  Small 
m_{cloud}  0.70  4.28  Large 
As was the case with Lurch 1 Linear, most of the variables (5 of 8) act to intensify the lurch. By far the largest effect is from the intensifier m_{cloud}. The sensitivities for most variables are the same as for Lurch 1 Linear, whereas the effects are greater.
Summary
Lurch 1 Angular is similar in most ways
to Lurch 1 Linear, except that it gives a rearward lurch that
is 35% faster than the linear version, which in turn is greater than the
observed. This means that Lurch 1 Angular is representing the physical
situation less faithfully than its linear analog. The sensitivities are also similar to Lurch 1 Linear,
with m_{cloud} being by far the most important.
Back to Lurch 7 Linear
Ahead to Lurch 2 Angular
Back to Physics of the Head Shot