6. The forward snaplinear calculations

This section addresses the first numerical question listed above, whether the observed forward snap of JFK's head (between 312 and 313) is quantitatively consistent with the impact of a WCC/MC bullet from the depository. To do that, we calculate the horizontal momentum of such a bullet, subtract from it the estimated momentum retained by the bullet as it exited from the head (in fragments), and set the difference equal to the momentum imparted to the head. From that, we calculate the forward speed given to the head and compare it to the speed calculated from the Zapruder film.
At the outset, we must be clear about exactly what speed we are calculating here and how it is to be compared with the speed observed from the Zapruder film. The equation given below calculates the final forward snap of the head, that is, the greatest speed it attains. Recall that as the bullet is passing through the head and imparting momentum to it, the head is gradually speeding up. As soon as the bullet exits the head, it can no longer transfer momentum to it, and the head continues to move at whatever speed it reached while the bullet was still inside. Thus the graph of speed of the snap vs. time will increase for a half-millisecond and then remain constant. The calculation gives that final speed.
The Zapruder film gives us something quite different, however. It yields an average speed over the entire frame between the close of 312 and the close of 313. Three separate factors make that speed less than the final speed as calculated by the equation. First, the time of the actual snap is less than a full frame because it almost certainly will start some time after 312 closes. Second, the time will be further decreased because the snap ended well before 313 closed (because 313 shows the explosion that followed the snap). Third, the distance covered by the snap will be greater than that shown in 313 because the explosion recorded there will have begun to push the head back (i.e., the head snapped farther forward than shown in 313).
Mathematically, these effects amount to:

In other words, the calculated answer must exceed the "observed" 3.3 ft s-1. If the head were to explode right after the bullet leaves the head (which it probably wouldn't, because the head requires a few milliseconds to build up enough pressure to burst, whereas the bullet takes only a half-millisecond to pass through the head), the calculated snap would differ the most from the true snap because the head would have the shortest amount of time to move forward. The longer the burst is delayed, the farther forward the head can snap and the more closely the actual snap will resemble the calculated one. Although it is theoretically possible to use the difference between the calculated and actual snaps to estimate how long the burst was delayed, it is very difficult to do it in practice because you also have to know how far back the head lurched. We content ourselves in simply recognizing that the calculated snap must exceed the actual snap, and possibly by a large amount.

The equation for the speed of the head snap
Applying the conservation of momentum to the hit to JFK's head gives the following simple equation

where Q is the angle between the bullet's path and the horizontal plane of the limousine. This can be converted into the formula

where mbullet = 161 grains, vbullet = 1800 ft s-1, vbulletafter = 200 ft s-1mhead = 7 lb, and Q = 12° (as default values). The result is vheadafter = 5.13 ft s-1 for the forward snap, which exceeds the observed speed of 3.3 ft s-1 by half. In light of the above discussion, the calculated (peak) speed is consistent with the observed (average) speed.

Sensitivity analysis
One of the major tasks associated with answering this first question is to understand which variables affect the answer the most. We do this by using a classical "sensitivity analysis," which determines the "sensitivity" of the answer to changes in each of the variables. The results are shown in the table below, with standard conditions shown in boldface. The sensitivities can be interpreted as the fractional change in the snap relative to the fractional change in the variable. For example, a sensitivity of 0.1 means that a 10% change in the variable produces only a 1% change in the snap, whereas a sensitivity of 1 means that 10% produces 10%, 1% produces 1%, etc. A negative sign, as seen for three of the five variables, means that the effect (the snap) varies oppositely to the variable (for example, greater masses for the head produce smaller snaps). The five sensitivities fall into two groups: the two insensitive variables vbulletafter and Q (-0.13 and -0.05, respectively) and the three order-of-magnitude more sensitive variables mbullet, vbullet, and mhead (1.00, 1.12, -1.00).

Sensitivity analysis, Snap 1 linear

 mbullet vsnap vbullet vsnap vbulletafter vsnap mhead vsnap Q vsnap 156 4.969 1750 4.967 7 5.213 157 5.001 1760 4.999 0 5.785 5.0 7.179 8 5.200 158 5.032 1770 5.031 50 5.621 5.5 6.526 9 5.184 159 5.064 1780 5.064 100 5.456 6.0 5.983 10 5.167 160 5.096 1790 5.096 150 5.292 6.5 5.522 11 5.148 161 5.128 1800 5.128 200 5.128 7.0 5.128 12 5.128 162 5.160 1810 5.160 250 4.964 7.5 4.786 13 5.106 163 5.192 1820 5.192 300 4.799 8.0 4.487 14 5.081 164 5.223 1830 5.224 350 4.635 8.5 4.223 15 5.056 165 5.255 1840 5.256 400 4.471 9.0 3.988 16 5.028 166 5.287 1850 5.289 17 4.999 Sensitivity =  [(5.160-5.096)/2]/ [5.128/161] = 1.00 Sensitivity =  [(5.160-5.096)/20]/ [5.128/1800] = 1.12 Sensitivity =  [(4.964-5.292)/100]/ [5.128/200] = -0.13 Sensitivity =  [(4.786-5.522)/1.0]/ [5.128/7.0] = -1.00 Sensitivity =  [(5.106-5.148)/2]/ [5.128/12] = -0.05 Range = 0.32 Range = 0.32 Range = 1.31 Range = 3.19 Range = 0.20

Ordered summary of sensitivities

 Variable Sensitivity of vsnap Range of vsnap, ft s-1 Magnitude Positive effect on snap vbullet 1.12 0.32 Small mbullet 1.00 0.32 Small Negative effect on snap Q -0.05 0.20 Small vbulletafter -0.13 1.31 Medium mhead -1.00 3.19 Large

The speed of the snap is increased by increases in the mass of the bullet (mbullet) and the initial speed of the bullet (vbullet), both over a modest range (0.3 ft s-1). The speed of the snap is modestly decreased by increases in the angle of inclination Q  (range of 0.2 ft s-1), moderately decreased (range of 1.3 ft s-1) by the exit speed of the bullet (vbulletafter), and strongly decreased (range of 3.2 ft s-1) by the mass of the head (mhead). These effects can be understood as follows: Mass and velocity of the impacting bullet contribute directly to the snap by creating the momentum that is transferred to cause it. The angle of inclination Q works inversely because greater inclinations reduce the horizontal momentum that creates the horizontal snap. The exit speed of the bullet (vbulletafter) works inversely because the faster the bullet leaves, the less of its momentum has been transferred to the head to create the snap. Finally, the mass of the head works inversely on the snap because the heavier the head, the more momentum is required to create a given speed of snap.
We will try to intuitively understand as many of the sensitivities and ranges as we can, for it is important develop gut-level feelings for this physics. But that becomes harder as the number of equations and variables increase. We need to continue to try, however.

The most important variables
The importance of a variable to the snap is controlled by its proportional effect on the snap (its sensitivity) and the range over which the variable can be reasonably expected to vary. That variation produces a range in the snap that is the real thing we can focus on. The sensitivity and range of effect must be considered together. For example, a variable can be sensitive but not have a large effect if it doesn't vary enough. Conversely, a less sensitive variable could have a larger effect on the answer if it varies more widely. For example, consider the three sensitive variables in the table of ordered sensitivities above (vbullet, mbullet, mhead). Each has a sensitivity of about 1, but two of them (mbullet and vbullet) have small enough ranges that they can change the snap by only 0.3 ft s-1. But the third variable (mhead) is known less precisely, enough so that its range of values (410 lb) produces a tenfold larger range in the snap (3 ft s-1). Thus the mass of the head must be considered the most "important" variable for the snap. It is ironic that the most important variable here gains that status by our inability to estimate its value.