10. Lurch 2 linear—add large fragments

The model and its justification
    The next step in making the simulation more realistic is to add the two large fragments to it. These are the while lines seen in Z313 and to a small extent in Z314. Their masses can be estimated from the pictures in Lattimer's book and the density of bone. They come out to be 0.027 gr for fragment 1 (with the upward angle of 40° and 0.01 gr for the one with an angle of 70° (here called frags₂₃ because it may include a smaller third fragment as well). We use a default velocity for each fragment of 500 ft s^-1, as measured on Z313 (keeping in mind that the path represents less than a full frame's exposure). These velocities turn out to be unimportant.

Solving the simultaneous equations
    The equations to be solved simultaneously are shown below. Note that the mass of the lurching body has been reduced by the masses of the fragments. The starting values of the variables are shown below the equations.
    The solutions for v_bodyafter and v_cloud, -2.18 ft s^-1 and 388 ft s^-1, respectively, are shown below the variables. Both are very similar to the values from the first linear simulation, -2.17 and 425 ft s^-1.

Conservation of X-momentum

Conservation of total energy

Default values of variables

Default solutions to simultaneous equations

v_bodyafter = -2.18 ft s^-1  v_cloud = 388 ft s^-1

Distributions of momentum and energy

    The table of distributions is very similar to the one without the large fragments, particularly for the momentum. The fragments removed 144 ft-lb of KE from the cloud, however (17% of its value). But the lurch changed by only 1%. Thus adding the fragments didn't change the basic solution very much.

Sensitivity analysis
    Although the sensitivity analysis looks different from the previous one, its essentials are the same. None of the six new variables (two masses, two speeds, two angles) have important sensitivities or ranges.

Sensitivity analysis, Lurch 2 linear
(Standard conditions in boldface)

Ordered summary of sensitivities

    None of the six new variables (associated with the mass, velocity, and angle of inclination of the two large fragments) affect the speed of the lurch very much, as seen above. They all have low sensitivities and small ranges of effects. Five of them have positive effects, one negative. The effects of the previous variables remain much the same. In other words, the large fragments are not important to the solution, at least for these early calculations.

Summary
    The answer with the two large fragments is fundamentally the same as without them. They are thus minor variables. This is confirmed by the six sensitivities and effects associated with them, which are all very small. The next step is to add three-dimensional motions, beginning with the diffuse cloud of fragments.

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