24. Introduction to ranges and constraints

    The major task of the previous sections has been to identify the most important variables out of the 30 or so that are used in the most complex simulations. The results were listed in two tables in section 23, "Summary of the solutions and the most important variables." Those variables and the range of solutions derived with their default values show clearly that the more realistic the simulation, the more the answers approach the observed rearward lurch. This answers the second question, Can the rearward lurch be explained by a shot from Oswald's rifle?, in the positive.
    The next step is to see whether the lurch must be rearward, in other words to address the third question, Must a rearward shot from Oswald's rifle have created a lurch similar to that observed? We do this by examining all reasonable combinations of the important variables and seeing what the results imply for the velocity of the lurch, the factors that control it, and the uncertainties in these answers. The answer turns out to be positive, that all reasonable combinations of the important variables produce rearward lurches compatible with the actual one. In fact, the majority of the reasonable combinations produce initial lurches that are stronger than the actual one, making the practical problem how to weaken the lurch rather than how to strengthen it. In other words, there must be a rearward mechanical lurch, and it must be large enough to be easily observable. This solves the mystery of the lurch—it begins with a quick mechanical recoil from the cloud and large fragments thrown forward, and continues with a more modest acceleration that cannot have been from a bullet (from any direction) because it is too prolonged. This leaves no room for a frontal hit.

Constraints
    The key to narrowing down the range of the lurch and other variables is to take advantage of several built-in constraints on the important variables. Any answer that violates one or more of these constraints can be rejected. It must be recognized, however, that only some of the variables are constrained. Here is a list of the major constraints:

1. The mass of the cloud expelled from the head (m_cloud) cannot be negative, zero, or exceed something like 1 lb. In other words, 0 < m_cloud < 1 lb.
2. The velocity of the cloud cannot exceed the speed of sound, which is about 1100 ft s^-1. That would require too much energy. The velocity also cannot be zero or negative. These constraints can be summarized as 0 < v_cloud < 1100 ft s^-1.
3. The velocity of the large fragments (v_frags) must be positive because they were seen to move forward (along the X-axis defined by the plane of the Z-film). It also cannot exceed the speed of sound, which is about 1100 ft s^-1, because as with the cloud, that would require too much energy. The velocity cannot be zero, since they were seen to move. These constraints can be summarized as 0 < v_frags < 1100 ft s^-1.
4. The potential energy (KE used to break the head) cannot be zero or negative. It also cannot exceed roughly 600 ft-lb, a value far in excess of the 200–300 ft-lb required to break the head. Lastly, it cannot be positive but less than about 100 ft-lb, a value well below the 200–300 ft-lb required to break the head. These constraints can be summarized as 100 ft-lb < PE < 600 ft-lb. The most likely values will probably be 200–400 ft-lb.
5. The distance of the snap, d_snap, must exceed the measured 2.2 in because the head had exploded and was moving backward by the end of frame 313. The distance must also be positive, since the head snapped forward. In other words, d_snap > 2.2 in.
6. The speed of the snap is constrained similarly to the distance of the snap. The speed must exceed the average of 3.3 ft s^-1 calculated from the full frame of 312 to 313. In other words, v_snap > 3.3 ft s^-1.
7. The radius of the head, R_head, cannot be be significantly less than that of an adult male head. That means roughly R_head > 3 in.
8. The mass of the head, m_head, cannot be less than 4 lb, the weight of the brain (1500 g) plus a minimal allowance for the skull. It probably can't be less than 5 lb.
9. The time taken by the lurch in frame 313, t_lurch, must be less than the time that the frame is open, 1/40 s, because the lurch began after 313 opened and ended before it closed. In other words, t_lurch < 25 ms.

    Some of these constraints are stronger than others. The strongest constraints are the absolute ones, such as that the potential energy cannot be zero or negative (1). The less-strong constraints usually involve a value that seems obvious but is not absolute. An example would be that the mass of the head cannot be less than 4 lb (8). Most of these nonabsolute constraints are actually stronger than they first appear. It is for this reason that we do not differentiate among them in practice.

Dealing with the important variables
    One effective way to search for constraints is to vary one or two of the important variables and list their effects on other variables in a series of tables. A solution disallowed for one variable must then be disallowed for all, which means that the same pattern of constraints must be applied to every variable. In other words, an allowable solution may not violate the constraints on any of the variables.
    Given the number of variables and constraints listed above, the process of searching out all allowable combinations is daunting. It can be simplified, however, by breaking the procedure into two steps, a prescreening (preconstraining) followed by a more-detailed constraining. This is made possible by the fact that two of the important constraining variables, t_lurch and v_frags, combine with m_head and d_snap to form a group of four that is related to the snap and is nearly independent of the variables more directly related to the lurch, such as Θ_cl, m_cloud, v_cloud, and PE. This first group can be used to find the allowable combinations of m_head and d_snap, the lurches for which can then be examined in more detail.

25. Preconstraining with m_head, d_snap, t_lurch, and v_frags
26. Constraints on Θ_cl and PE
27. Constraints on m_cloud and v_cloud from Θ_cl and m_cloud
28. Constraints on m_cloud and v_cloud from Θ_cl and PE
29. Constraints on m_cloud and v_cloud from m_cloud and v_cloud
30. Grand summary of constraints

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