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 (mcloud) cannot be negative, zero, or exceed something like 1 lb. In other words, 0 < mcloud < 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 < vcloud < 1100 ft s-1.
3. The velocity of the large fragments (vfrags) 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 < vfrags < 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, dsnap, 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, dsnap > 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, vsnap > 3.3 ft s-1.