34. Scenario 3—frontal hit immediately after rear hit

This third scenario is the overt combination of a rear hit that does everything followed immediately by a frontal hit that adds only momentum (frontal scenario 1). This is the version proposed by Josiah Thompson in Six Seconds in Dallas. The rear hit snaps the head forward, then explodes the head and creates the cloud and the two large fragments. Its proponents do not claim that it also creates the lurch, but this monograph shows that it must create a sizeable lurch. The frontal hit then simply adds momentum to the lurch. The question considered here is what happens when these two scenarios are combined into a single set of equations, not just considered separately. The answer is that the two effects combine linearly, and so can continue to be considered separately for simplicity.

The procedure for calculating
Scenario SL6L was chosen for simplicity, so as to avoid having to deal with the tricky Rhead and Rbullet of SN6A. To the equations were added terms for a frontal bullet, mbullet2, vbullet2, vbulletafter2, Θv2, and Θh2. Default setting were as before: mhead = 6 lb, Θv = 12°, Θh = 9°, mcloud = 0.3 lb, dsnap = 2.4 in, and PE = 300 ft-lb. The first bullet (from the rear) was taken to be a Mannlicher-Carcano, with mbullet = 161 gr and vbullet = 1800 ft s-1. The second bullet (from the front) was taken to be a Remington .221 for the XP-100 Fireball, with mbullet2 = 50 gr, vbullet2 = 2650 ft s-1, and vbulletafter2 = -200 ft s-1.
The modified conservation equations are shown below. Since the second bullet was not part of the explosion caused by the first bullet, it did not participate in it directly. That means that it only had to added to the equation for conservation of momentum.

##### Conservation of total energy

The results
The results of using these new equations can be expressed very simply, for the equations represent the combination of scenario SL6L with new scenario 1 for the frontal hit. We test whether the joint equations give the same answers as the equations used separately. If they do, the extra lurch from the frontal shot can just be added to the lurch from the rear shot, and there is no more need to use these combined equations.
The table below shows the results. The lurches are strictly additive, and the velocities of the cloud are additive by default, since the frontal hit does not contribute to the lurch from the rear hit. These results mean that these two scenarios add to one another, and can be considered separately. The rear hit creates a rearward lurch of 1.05 ft s-1. The frontal hit creates a rearward lurch of 0.17 ft s-1 by itself. The two bullets combined create a lurch of 1.22 ft s-1, which is just the sum of 1.05 and 0.17.

 Scenario vlurch, ft s-1 vcloud, ft s-1 Rear hit from MC -1.05 406 Frontal hit from XP-100 -0.17 — Rear and frontal hits together -1.22 406