CHAPTER 2 INITIAL COLLISIONS OF PRIMARY PARTICLES
[Note: For this chapter it
might be useful to have a couple of full size pencils handy to help
visualize what is being described.]
Section 1: Infinite
quantities and rod motion
What length is this rod? What units should we use? Let’s consider using a dimension-less quantity, that is a ratio: width to length. Taking the width to be = 1 part, then the length to width ratio will = 1:?. The value for length is of no real concern yet, but in Chapter 5 it is postulated to be 137.
If two cylindrical particles A & B are brought together with their tops being both in the same plane, there is then perfect alignment. For 2 particles traveling randomly in space what is the chance they would collide in perfect alignment? Near zero. Why? By Principle 10 there are an infinite number of misaligned positions; therefore A-B can “never” be perfectly aligned by random collision. However if never then that denies that the contact occurs across given points, but what might be supposed is for any given alignment the chance of such through random collision is 1/infinity.
Another way to state the same is for any difference in alignment, that difference is infinitely sub-dividable, so perfect alignment is infinitely impossible (or not possible, except for 1/infinity).
Consider the motion of a rod (B) as perpendicular to its ends. If it hypothetically hits a rod A (stationary) at the midpoint of each rod (that 1/infinity chance), and perpendicular to A for discussion sake, then;
Principle 12: the speed of
motion times the quantity of rod above and below the point of contact
being equal, there would be no preference for rotation to occur at
either arm so consider rod A accelerated in a linear
direction as per Principal 9.
Principle 13: for any speed
of motion times the quantity of rod above or below the point of
contact, this being unequal, the rod will rotate in the direction of
the longer arm.
motion of the rod so involved goes to a rotational quality. This
rotational motion is as “natural” as linear and itself can be
transferred rod to rod (see Case #3, Chapter 3). So:
Principle 14: Rotational motion continues until the rod rotating reaches its mid-point and is at balance, and then linear motion is re-formed.
When the rotating rod, progressing around rod A reaches its midpoint, its arms being balanced, it returns to a linear motion again, as per Principle 12, and so accelerates A (now in a linear direction that is perpendicular from the ends of B at the point where it has reached its midpoint).
the rods may be aligned (perpendicular to each other) at their
midpoints by the process of rotation as this passes one point across
another, which is a different sense from that of random possibly in
Transfer of Motion Rod to Rod
Any tangent drawn to a perfect circle contacts the circle but no area can be described for the point of contact. As, for any chord drawn parallel to the tangent, consider the arc between the chord and tangent. For any chord drawn further toward the tangent (still parallel) this arc only gets smaller but an arc remains ad infinitum. Such a point on a circle is of a mathematical point quality (indefinable by area). Therefore the point of contact of two curvilinear figures is a point of “zero” area, to which motion must pass through this “window” to be transferred rod to rod (See Appendix B for more information on this).
At one time I considered that this motion across the mathematical point(s)needed to be in a straight line. Then I considered any angle is possible across the window of contact, short of parallel to the rods. Now I consider this; force toward window means only if constrained along short axis will it torque. The point of contact on a rod is determined by that axis on the circular, that is the diameter. It is not a long rectangle with rounded ends, nor an elongated oval. In which case the long axis would relate to the contact point also, and the window would cut wedges for the window opening. But for a rod it cuts one circular cross section across rod only as it is only that axis that relates to the circular nature of the rod, and therefore the “style” of the mathematical point of contact (you have a whole line on the other/long axis). But when this motion comes across it radiates/spreads out to the full area. For if it filled across in a straight line, and dragged the particle into motion by infinite friction between mass points, then the notion of rotation has no basis. Instead I see this scenario
1. Motion within a rod without contact is as is. This is primary motion type.
2. Motion on impact within that rod is to the point of contact with respect to a circular cross section cut in that rod (perpendicular to its ends). This then is either symmetrical (resulting in a linear push) or unsymmetrical resulting in a rotational push/motion. This is secondary motion type.
That is when this movement of motion, expressed with vectors, is symmetrical with respect to this/these contact points, then linear motion occurs and/or is transferred.
3. Motion across the window of contact to the other rod is as to a circular cross section cut on the other rod (perpendicular to its ends) and brought back though the impacting rod. This means any movement though that window of contact causes pressures related to movement to that cross section and the related point of contact. Such pressure/motion vectors are what will torque the rod. This is the tertiary type of motion.
4. After crossing the window of contact the motion fans out/radiates. This is the fourth type/level of motion, with the resulting pressures taking that rod “back” to a level one motion.
5. The vectors of motion then if symerical to a circular cross section that is perpendicular to the ends of the rod, and drawn from the radiation point, will result in linear motion. Else if there is an imbalance the rod rotates forward in relation to the average motion vector, "pivoting" virtually from the radiation point (though it is on the backside to the direction of rotation).
6. This causes a back swing which hits the first particle and causes motion to reverse to that point, leading to a change of rotation onto the first rod.
This would mean
7. The pre-dominant motion of any movements, be it a total or of an excess quality and quantity in a rod, reforms that portion of motion to itself
So, these four “levels” of motion are related to certain circumstances; free movement, impacts, the window of contact, filling new mass area, completion of the process.
8. Any type of impact that presents a situation where motion cannot be expressed in space as it had been, results in transfers of motion, torque, rotation or rebound, or vibration as a last resort, to satisfy the conservation of motion principle #4 and occurs along the lines of the other principles so mentioned here and elsewhere (in the case of rebound and vibration).
9. Impacts cause a movement of motion to the point of impact, and in so doing rods can reverse direction, either linearly or rotationally, as well as undergo torque when not perpendicular.
If the rods are traveling though space and collide, by principle 10 the rod (B) would be slanted relative to the hit rod (A).
As above (principle 15.1.3) Motion across the window of contact to the other rod is as to a circular cross section cut on the other rod (perpendicular to its ends) and brought back though the other rod. This means any movement though that window of contact causes pressures related to movement to that cross section and the related point of contact. Such pressure/motion vectors are what will torque the rod. This torques the rod to the point the sides are equal, then there is as symmetry that rests the rod in that position, pressures being balanced on that "axis".So a force of motion to torque the rod B perpendicular with rod A would occur, until the perpendicular position is reached and forces balance.
This torque occurs as a motion in space that takes time, and before any other acceleration can happen, but occurs at the same time a unison motion component occurs between A and B, only the excess over this goes into torque.
Now there are fundamentally two types of collisions, direct hits and overtaking hits, gone over in detail in Section 2. Suffice to say as an introduction, on direct hits all (absolute) motion is stopped, goes into torque per principle 15.1.3, then rotation per principle 13, then back to linear motion per principle 14. For an overtaking hit there is a unison motion that is achieved by the overtaking particle (B) with the overtaken particle (A), thereafter any access above the unison motion goes into torque and then rotation, The unison motion is initially following the the same vector of motion B had traveling to meet A, therefore in theory B would slide across A relative to it.
The excess motion, after torque, means any excess motion as rotation is in a perpendicular to
In the case of B sliding by A due to the vectors at angles to each other, this unison motion establishes and occurs while B is torqueing.
Then after rotation the added problem is with the direction of the linear motion of A and B different, the rotation causes an additional displacement of B relative to A that is a change from the established unison motion at the first instance. That is the rod B itself has changed position, so that its motion though space, unless the same as A, intersects A’s linear motion at different spots, even though B’s linear motion is in the same direction to the horizon as when established in the first instance. This can be seen most clearly when B comes to the top of A, it is now clearly moving up and off of A. Conversely if rotating to the bottom of A, all its linear motion is suppressed.
HOWEVER, this doesn't occur because in the first instance as B begins to rotate it "tests" the window of contact as it were with linear motion which combines with the linear unison motion and because it diminishes toward the linear motion of A that would , as it where, increase rotation and the testing cycle occurs again. So all B's linear overtaking motion moves toward a equal plane (but not line) of direction as A's linear motion, increasing the amount of rotation possible as it does. This change of direction of B's linear motion is instantaneous as all occurs within the rods mass.
After which B actually begins rotating on A, with a unison motion component now in the same plane as A's linear motion. Now each point of rotation causes a testing which also vectorizes with the unison motion of B.
If the long arm of rotation is toward the back of A (behind the direction of A's motion) it causes an increase in rotation and more vectorization, to a limit of what is necessary for a new unison motion keeping up with A. All these times then B is sliding relative to A as well as rotating. Only when B rides across the point in the back of A, that is behind its vector of motion, does the unison motion of B comes in equal to that of A, having been drawn down from the original , and each step of thereafter, of the unison motion B had.
After which A and B move off with unison speed and direction, and with B rotating on A. Now each point of rotation causes a testing but the rotational motions "linear test" does not vectorize with the unison motion thence because it would need a loss of rotational motion, which cannot occur unless the arms of B are balanced. So several principles are needed here:
15.4 A linear "testing" occurs at each point of rotation
of a rod rotating on another rod. For overtaking hits this
instantaneously forms a resultant with its unison overtaking motion.
If this secondary linear motion can be expressed, in the rod or by
accelerating another rod, then the resultant with the unison linear
motions are compounded (two simultaneous linear motions occurring within rod).
If this secondary linear motion is impended, as it were, before (and
causing) rotation, the two linear motions merge to the resultant,
forming one linear motion.
Principle 15.5 If, in the
resultant impending motion of the overtaking rod, motion can be lost
from the unison motion component to an increase in rotation, to make
the resultant not impending, this occurs.
Principle 15.6 If in the
resultant impending motion of the overtaking rod, and motion must be
taken from the rotational component, this does not occur, unless the
rod is at midpoint balance, but unison linear motion toward a horizon
remains as it is, likewise the rotational motion remains at the same
As in the case described before Principle 15.4
Section 2: Collisions
of Two Particles
Now, in more detail,
to consider how two particles might normally collide, that is off
center to each other. There are two basic ways these two particles
might collide, via a direct hit, or by one overtaking another.
1. B could overtake A from
2. Also with many “slants”
relative to A.
3. To the extent B has the
velocity to overtake A, there is a unison motion component of B,
following its original motion vector, which would slide B across A.
4. Any excess velocity then
goes into torque of of B.
5. After torque this excess
velocity over the unison notion goes into rotation of B on A, on the
perpendicular cross section of A.
6. However (if the long arm of rotation is toward the back of A) at first instance
B testing on A causes all B's unison motion to swing in line with A's
linear motion as per principle 15.4.2 and 15.5
B rotates on A, while they both maintain a unison linear motion
toward the same horizon per principle 15.6.
When B reaches its midpoint, linear motion is renewed per principle
14, and B accelerates A, such that A rotates back around B (principle 15.1.6), while
they are still in unison motion as in (7),
When A reaches its midpoint A and B are at midpoint to midpoint
a) If this is on the "overtaking side" of B then that rotational motion, now all in A reverts to linear accelerating B as per principle 9. And they go off in unison with a compounded linear motion as per principle 15.4.1 This means on overtaking hits doublets are formed and the motion of the same will be in a slightly different direction than the primary particle flow.
b) If on the direct hit side there is a problem with the transfer of linear motion from A to B. It is contrary to the unison linear motion of both B and A such that the rods would be having two opposite (not overlapping here) vectors of motion. In this case of contrary linear motions in A it causes the contrary motion to be draw together with the unison motion as one vector. However not as standard vector analysis either, but the drawn vector is in the spot as triangulated, but with the length of each vector added (see Appendix A, Section 4 Contrary Motion Vectors, especially principle 19.2). So the Rods A and B go off together with the same speed and direction, but the vector of motion relative to the ends of the rods may be different.
Proceeding from 5. If the long arm of rotation is toward the front of A, then B's unison motion, indeed all B's linear motion, diminishes to zero, When and if B rotates in front of A treat as a direct hit.
With a direct hit A-B, by Principle 10, both A and B will be off midpoints. All linear motion of each goes to torque and then rotation of each other.
One PP (A) will reach its midpoint before the other (according to their relative velocity’s and initial positions) where it will then transfer all its velocity into the (added) rotation of the other PP (A becomes stationary).
When that PP reaches its midpoint (both PP’s then are at a midpoint – midpoint contact) it will move in a linear direction moving each equal to ½ the velocity of its last rotational speed (by Principle 9).
So on the overtaking hits the result will almost always (unless B slides by A during torque) be two particles moving off in unison in the direction of the field of the PP flow*, but with directions divergent somewhat from the original motion of the particles. Or with two particles doing the same but with a wider divergence from the original motion of the overtaken particle. A direct hit will result in two particles moving off in unison but randomly in any direction.
* (except if one got in to more extended case studies of reverse overtaking from PP flow)