26. Constraints on Θcl and PE

This section begins our consideration of the four variables most important to the lurchΘcl, mcl, vcloud, and PE (the obvious vcloud was not on the original list because it was solved for rather than set). Note that three of the four have to do with the cloud. This makes them difficult to estimate, and requires that we use ranges for each and look for constraints.
We begin with potential energy (PE) because it is the easiest to handle. It is independent of the others and has an effective range of 100–600 ft-lb, as discussed in the Introduction to Constraints. We will use this range later to help find the constraints on other variables.
In contrast to PE, Θcl is not mathematically or physically limited other than by the obvious 0° and 180° (pencil-thin cloud pointing straight ahead and spherical cloud, respectively). In practice, we can say that its lower limit must have been wide enough to cover the front part of the car with debris, which would make it something like 10°–15° (half-angle) to span the the front corners of the car and 30° to span the front of the passenger compartment (in front of the Connallys). Of these, the 30° seems more reasonable. Its upper limit would be the value that just prevented debris from reaching the trunk, where none was reported. To avoid the entire trunk would take a half-angle of 135° (because the trunk subtended an angle of 90° from Kennedy's head). This makes the effective range of Θcl something like 30°–135°. As with PE, however, the range of likely values would be smaller, possibly something like 50°–60° to 90° or so. Recall that we use 70° as default value.
We can now examine how Θcl affects other variables within its range. The easiest way to begin is to allow Θcl to vary and list the effects on other variables. This is done in the table below, based on calculations with SL6A and default values for the other variables (in particular mcloud = 0.3 lb, PE = 300 ft-lb, and mhead = 7 lb).

 Θcl, degrees → 10 20 30 40 50 60 70 80 90 100 110 120 vlurch, ft s-1  ↑ -2.85 -2.74 -2.57 -2.33 -2.05 -1.73 -1.39 -1.04 -0.70 -0.38 -0.09 0.16 vcloud, ft s-1  ↑ 384.64 384.74 384.89 385.08 385.28 385.48 385.65 385.78 385.88 385.94 385.96 385.96 Ωbodyafter   ↑ -8.38 -8.06 -7.54 -6.85 -6.02 -5.08 -4.09 -3.07 -2.07 -1.13 -0.27 0.47 Ωcloud  ↓ 10.71 10.39 9.87 9.18 8.35 7.42 6.42 5.40 4.40 3.46 2.60 1.86 Sum 2.33 2.33 2.33 2.33 2.33 2.34 2.33 2.33 2.33 2.33 2.33 2.33 KEbodyafter  ↓ 4.78 4.42 3.87 3.20 2.47 1.76 1.14 0.64 0.29 0.09 0.01 0.02 KEcl   ↑ 693.50 693.86 694.41 695.08 695.81 696.52 697.14 697.64 697.99 698.20 698.28 698.27 Sum 698.28 698.28 698.28 698.28 698.28 698.28 698.28 698.28 698.28 698.29 698.29 698.29

First the basic observations from the table:

1. Both vlurch and vcloud increase as Θcl increases (as shown by their up-arrows). See the discussion after this list for more details.
2. The angular momentum of the lurch, Ωbodyafter, increases with increasing Θcl, while Ωcloud  decreases. Their sum remains constant, however, as required by conservation of momentum. (See graph and discussion below for more details.)
3. The two angular momentums are of comparable size, whereas the two kinetic energies (lurch and cloud) are notKEcl >> KEbody.
4. The KE of the lurch, KEbodyafter, decreases with increasing Θcl, whereas the KE of the cloud, KEcl, increases. As required by the conservation of total energy, the sum of KEbody and KEcloud remains constant. (See graph below.)
5. Since KEcl >> KEbody, KEcl changes by a small fraction, whereas KEbody changes by a large fraction.
6. KEcl is not a significant contributor to the total KE. It essentially "floats" in response to changes in the cloud. As Θcl increases, vcloud and KEcl increase, and KEbody decreases to balance KEcl.

This series of observations is not intuitively obvious, however, and needs further explanation. We address it in detail because it offers insight into the underlying physics of JFK's movements.
Consider point 1 above, that
velocity of the both lurch and the cloud increase as Θcl increases. On the face of it, this should not be, since they are supposed to act in opposite directions—since the lurch is a recoil from the cloud, it should increase (in speed) when the cloud increases in speed. Instead, we see the reverse happening here—the speed of the cloud strengthens while the speed of the lurch weakens. How can this be?
The solution involves the conservation laws and the X-component of the cloud. First note that it is the average X-component of the cloud's momentum (or angular momentum, as here) that must balance the momentum in the lurch (excluding the for moment smaller terms like the fragments), not just the speed of the cloud itself. [The speed vcloud is the same in all directions, but the part that moves up-down or in-out (of the plane of the film) contributes less to the X-velocity.] We transform vcloud into its X-component by multiplying by fxcl according to the formula given earlier.
Now imagine what happens as the cloud of fragments broadens (Θcl increases) while the speed of the fragments (vcloud) remains the same. The forward component of the the speed will decrease, and the rearward lurch will decrease to keep the total forward and backward momentum constant. This balances the momentum, but not the KE, for now we have a decrease in KE of the lurch but not of the cloud (because KE uses total speed, not a directional component). The total KE has decreased, which is not allowed to. In order to preserve the total KE, the speed of the cloud (vcloud) must increase while preserving its decrease in the X-direction. The fxcl term takes care of that. So X-momentum is preserved by decreasing the rearward speed of the lurch, while KE is preserved by slightly increasing the speed of the cloud to compensate for the decreased KE in the lurch.