**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 lurchit 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:

- The mass of the cloud expelled from the head (
*m*) cannot be negative, zero, or exceed something like 1 lb. In other words, 0 <_{cloud}*m*< 1 lb._{cloud} - 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*< 1100 ft s_{cloud}^{-1}. - The velocity of the large
fragments (
*v*) 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_{frags}^{-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*< 1100 ft s_{frags}^{-1}. - 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 200300 ft-lb required to break the head.
Lastly, it cannot be positive but less than about 100 ft-lb, a value well below the 200300 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 200400 ft-lb. - The distance of the snap,
*d*, 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,_{snap}*d*> 2.2 in._{snap} - 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*> 3.3 ft s_{snap}^{-1}. - The radius of the head,
*R*, cannot be be significantly less than that of an adult male head. That means roughly_{head}*R*> 3 in._{head} - The mass of the head,
*m*, 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._{head} - The time taken by the lurch in frame 313,
*t*, 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,_{lurch}*t*< 25 ms._{lurch}

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

25.
Preconstraining with *m _{head}*,

26. Constraints on

27. Constraints on

28. Constraints on

29. Constraints on

30. Grand summary of constraints

Back to Summary of Solutions and Variables

Back to Physics of the Head Snap