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.

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