26. Constraints on Θ_cl and PE

    This section begins our consideration of the four variables most important to the lurch—Θ_cl, m_cl, v_cloud, and PE (the obvious v_cloud 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, Θ_cl 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 Θ_cl 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 Θ_cl affects other variables within its range. The easiest way to begin is to allow Θ_cl 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 m_cloud = 0.3 lb, PE = 300 ft-lb, and m_head = 7 lb).

    First the basic observations from the table:

1. Both v_lurch and v_cloud increase as Θ_cl increases (as shown by their up-arrows). See the discussion after this list for more details.
2. The angular momentum of the lurch, Ω_bodyafter, increases with increasing Θ_cl, while Ω_cloud  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 not—KE_cl >> KE_body.
4. The KE of the lurch, KE_bodyafter, decreases with increasing Θ_cl, whereas the KE of the cloud, KE_cl, increases. As required by the conservation of total energy, the sum of KE_body and KE_cloud remains constant. (See graph below.)
5. Since KE_cl >> KE_body, KE_cl changes by a small fraction, whereas KE_body changes by a large fraction.
6. KE_cl is not a significant contributor to the total KE. It essentially "floats" in response to changes in the cloud. As Θ_cl increases, v_cloud and KE_cl increase, and KE_body decreases to balance KE_cl.

    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 Θ_cl 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 v_cloud 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 v_cloud into its X-component by multiplying by f_xcl according to the formula given earlier.
    Now imagine what happens as the cloud of fragments broadens (Θ_cl increases) while the speed of the fragments (v_cloud) 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 (v_cloud) must increase while preserving its decrease in the X-direction. The f_xcl 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.

Ahead to Constraints on m_cloud and v_cloud from Θ_cl and m_cloud
Back to Preconstraints
Back to Intro to Constraints
Back to Physics of the Head Shot