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:

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.

Conservation of X-momentum

Conservation of total energy

Default values of variables

Solutions to simultaneous equations

v_bodyafter = -2.92 ft s^-1  v_cloud = 425 ft s^-1

Distributions of momentum and energy

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)

Ordered summary of sensitivities

    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.

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