16. Lurch 1 Angularwith 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 = 0.5mv2 KE = 0.5mv2fI/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 fI is a geometrical factor having to do with the shape of the object that is rotating (fI 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 vbodyafter 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.

##### Conservation of total energy

Default values of variables

 mbullet = 161 gr vbullet = 1800 ft s-1 mbody = 85 lb vbulletafter = 200 ft s-1 mcloud = 0.3 lb PE = 300 ft-lb Q = 12° fI = 1.11

### Solutions to simultaneous equations

vbodyafter = -2.92 ft s-1  vcloud = 425 ft s-1

## Distributions of momentum and energy

 Momentum Energy, ft-lb Before After Before After Ωbullet = 3.80 Ωbulletafter = 0.43 KEbullet = 1164 KEbulletafter = 14 Ωbodyafter = -8.58 KEbodyafter = 4 Ωcloud = 11.95 KEcloud = 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 mcloud, followed at a distance by mbody 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)

 mbullet vlurch mbody vlurch mcloud vlurch vbullet vlurch vbulletafter vlurch 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 =  [(-2.931 + 2.911)/2]/ [2.921/161] = -0.55 Sensitivity =  [(-2.759 + 3.104)/10]/ [2.921/85] = 1.00 Sensitivity =  [(-3.549 + 2.176)/0.2]/ [2.921/0.3] = -0.70 Sensitivity =  [(-2.945 + 2.897)/20]/ [2.921/1800] = -1.48 Sensitivity =  [(-2.938 + 2.900)/100]/ [2.921/200] = -0.03 Range = 0.10 Range = 1.46 Range = 4.28 Range = 0.24 Range = 0.15

 PE vlurch Q vlurch fI vlurch 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 =  [(-2.800 + 3.039)/100]/ [2.921/300] = 0.24 Sensitivity =  [(-2.931 + 2.912)/4]/ [2.921/12] = -0.02 Sensitivity =  [(-2.895 + 2.947)/0.02]/ [2.921/1.11] = 0.99 Range = 1.46 Range = 0.05 Range = 0.26

Ordered summary of sensitivities

 Variable Sensitivity of vlurch Range of vlurch, ft s-1 Magnitude Positive effect on lurch (reduces rearward velocity) fI 0.99 0.26 Small PE 0.24 1.46 Large mbody 1.00 1.46 Large Negative effect on lurch (increases rearward velocity) Q -0.02 0.05 Small mbullet -0.55 0.10 Small vbulletafter -0.03 0.15 Small vbullet -1.48 0.24 Small mcloud -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 mcloud. 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 mcloud being by far the most important.