9. Lurch 1 Linearthe simplest analysis, with bullet, body, and cloud

The model and its justification
We begin the linear calculations of the lurch by using the simplest representation (model), in which the speed of the rearward lurch is calculated from only the the bullet, the body, and the diffuse cloud of fragments. The idea is that the impact of the bullet supplies momentum and energy that afterwards appear as a movement of the cloud forward and, in response, a movement of the body rearward. The variables involved here are mass of bullet, initial and final speeds of bullet, mass of body (torso), mass and velocity of cloud of fragments, angle of incoming bullet, and potential energy (kinetic energy used up in penetrating the skull on both sides). For simplicity, we assume that the cloud moves horizontally forward. The two variables that we solve for are the speed of the lurch (vbodyafter) and the speed of the cloud (vcloud). All the rest are either known or guessed at.

Solving the simultaneous equations
The conservation equations for momentum and energy that are to be solved simultaneously are shown below. Note that even though they are called masses, the "m" terms are now actually weights, and are divided by 32 to make the proper masses. Starting values for the seven variables are shown below them. The solutions for vbodyafter and vcloud, -2.17 ft s-1 and 425 ft s-1, respectively are shown below the variables. Note that again we have an embarrassment of riches, with the basic solution for the speed of the rearward lurch well exceeding the observed value. This must mean that this simple representation is not good enough.

Conservation of X-momentum

Conservation of total energy

Default values of the 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 Θ = 12°

### Default solutions to the simultaneous equations

vbodyafter = -2.17 ft s-1  vcloud = 425 ft s-1

## Distributions of momentum and energy

 Momentum Energy (ft-lb) Before After Before After Pbullet = 1.26 Pbulletafter = 0.14 KEbullet = 1164 KEbulletafter = 14 Pbodyafter = -2.87 KEbodyafter = 2 Pcloud = 3.99 KEcloud = 848 PE = 300

It is interesting to use the table above to see where the bullet's momentum and kinetic energy go. The original momentum, 1.26 lb ft s-1, is divided into positive and negative components (forward and backward) of 3.99 for the cloud. -2.87 for the body, and 0.14 for the bullet, assuming the default exited speed of 200 ft s-1. The energy divides itself very differently. The original 1164 ft-lb goes overwhelmingly to the cloud (848 ft-lb), with 300 going into potential energy (by default) and meager 14 and 2 to the exiting bullet and the body, respectively. To put it another way, the lurching body gets nearly half the momentum but less than 1% of the energy. The cloud get 98% of the final kinetic energy and 73% of the total energy.

Sensitivity analysis
The sensitivity analysis is shown below. It reveals that the most sensitive variables are mbody and vbullet. But the greatest importance (range of lurch) goes to mcloud, with its range of 3.2 ft s-1 for the lurch, far greater than from any other variable. This puts these calculations in a difficult light, for the variable with the biggest effect is again the one we know the least about (hard to reliably estimate the mass of the cloud).

Sensitivity tests, Lurch 1 linear
(Standard conditions in boldface)

 mbullet vlurch mbody vlurch mcloud vlurch vbullet vlurch vbulletafter vlurch 156 -2.128 1750 -2.077 157 -2.136 65 -2.835 1760 -2.095 0 -2.083 158 -2.144 70 -2.632 1770 -2.113 50 -2.109 159 -2.151 75 -2.456 0.1 -0.892 1780 -2.131 100 -2.131 160 -2.159 80 -2.302 0.2 -1.612 1790 -2.149 150 -2.150 161 -2.166 85 -2.166 0.3 -2.166 1800 -2.166 200 -2.166 162 -2.174 90 -2.046 0.4 -2.634 1810 -2.184 250 -2.179 163 -2.181 95 -1.938 0.5 -3.047 1820 -2.201 300 -2.188 164 -2.188 100 -1.841 0.6 -3.421 1830 -2.219 350 -2.194 165 -2.195 105 -1.753 0.7 -3.765 1840 -2.236 400 -2.197 166 -2.203 0.8 -4.087 1850 -2-254 Sensitivity =  [(-2.174 + 2.159)/2]/ [2.166/161] = -0.56 Sensitivity =  [(-2.046 + 2.302)/10]/ [2.166/85] = 1.00 Sensitivity =  [(-2.634 + 1.612)/0.2]/ [2.166/0.3] = -0.71 Sensitivity =  [(-2.184 + 2.149)/20]/ [2.166/1800] = -1.45 Sensitivity =  [(-2.179 + 2.150)/100]/ [2.166/200] = -0.03 Range = 0.08 Range = 1.08 Range = 3.20 Range = 0.18 Range = 0.11

 PE vlurch Q vlurch 0 -2.658 50 -2.581 7 -2.152 100 -2.502 8 -2.154 150 -2.421 9 -2.157 200 -2.338 10 -2.160 250 -2.253 11 -2.163 300 -2.166 12 -2.166 350 -2.076 13 -2.170 400 -1.983 14 -2.174 450 -1.888 15 -2.178 500 -1.788 16 -2.183 550 -1.685 17 -2.187 600 -1.577 Sensitivity =  [(-2.076 + 2.253)/100]/ [2.166/300] = 0.24 Sensitivity =  [(-2.174 + 2.160)/4]/ [2.166/12] = -0.02 Range = 1.08 Range = 0.04

Ordered summary of sensitivities

 Variable Sensitivity of vlurch Range of vlurch, ft s-1 Magnitude Positive effect on lurch (reduces rearward velocity) PE 0.24 1.08 Medium mbody 1.00 1.08 Medium Negative effect on lurch (increases rearward velocity) Q -0.02 0.04 Small mbullet -0.56 0.08 Small vbullet -1.45 0.18 Small vbulletafter -0.03 0.11 Small mcloud -0.71 3.20 Large

We must be very careful to note that positive sensitivities for the (rearward) lurch mean that increases in a driving variable increase the velocity of the lurch, which means making it less negative (because all the answers are negative, or rearward). This is counterintuitive, and must not be forgotten. Negative sensitivities mean that increases in the driving variable make the velocity of the lurch more negative, that is, increase its rearward component. In brief, positive sensitivities decrease the observed rearward lurch; negative sensitivities increase it. For simplicity, we will write "increase the lurch" when we mean strengthen it, or make it faster in the rearward direction, and "decrease the lurch" when we mean weaken the observed rearward motion.
That said, we can examine the table immediately above. Note first that five of the seven variables increase the lurch, and only two decrease it. Of the five strengtheners, only the mass of the cloud matters much; the mass of the bullet, its entrance and exit velocities, and its angle of impact have negligible effects on the lurch. Both of the weakeners, PE and mbody, have a moderate effect. These the mass of the could matters most to the final solution, and the mass of the body and the potential energy come in second. (This is the reason that Snap/Lurch 6 solves explicitly for the mass of the cloud.)

Summary
This first model for the lurch produces a value that significantly exceeds the observed value. A more sophisticated approach must be tried. The most important variables is the mass of the cloud, followed at a distance by the mass of the body and the potential energy.