A Non Fiction Trilogy

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).



Rotation


      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.


      All linear 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).

       So 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 collisions.


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, so I consider motion must pass through a “window” that only allows a perpendicular style transfer of motion. That is for motion to travel rod to rod it must travel across the mathematical point in a straight line perpendicular to the long axis of the rods. See Appendix B under Further Considerations for more information on this. So a principle; 


Principle 15: As two rods undergo an attempt to accelerate or actual acceleration, this occurs from rod to rod as perpendicular to the ends of the accelerating rod across the point of contact perpendicularly.



Torque


      After decades of moving ideas around as of December 2016 I have changed back to my "original" notion of torque occurring when two rods collide, and have archived my other main idea on causing perpendicular alignment at end of book. If the rods are traveling though space and collide, by principle 10 the rod (B) would be slanted relative to the hit rod (A). So indeed forces of motion within B, in order to pass a straight line needed by the constraint of the contact point A-B of a mathematical point, would tend to seek a level equal to a circular section cut from A and followed back into B as a elliptical section. The motion above this level, the upper arm of B, would have a downward component to the change in motion, and vice versa for the lower arm. 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 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. So


Principle 15.2: Upon impact, from direct or overtaking hits, the first change in motion of one or both particle is torque of the particle(s) to a position where they are exactly perpendicular to each other.


      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 15.2, 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, in passing though the window of contact, means any excess motion as rotation is in a perpendicular to A, therefore;


Principle 15.3 The excess motion in an overtaking particle, after torque, in passing though the window of contact, means any excess motion is as rotation in a perpendicular to the overtaken particle.


"Unison" Motion

        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 some 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, causing 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:



Principle 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.


15.4.1 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).


15.4.2 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 velocity.


      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.


Overtaking Hits


1. B could overtake A from many angles.


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, as per principle 15.7.


6. However 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


7. Thereafter B rotates on A, while they both maintain a unison linear motion toward the same horizon per principle 15.6.


8. When B reaches its midpoint, linear motion is renewed per principle 14, and B accelerates A, such that A rotates back around B, while they are still in unison motion as in (7),


9. When A reaches its midpoint A and B are at midpoint to midpoint contact.


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 (see Appendix A, Section 4 Contrary Motion Vectors) in A it causes the new linear motion to rebound, reverse as it is, and A moves off at a combined speed of the rotational speed, turned linear, and the original linear motion it had. Those two motions will be compounded and the result will be A traveling off away from B. Stated as a principle


Principle 15.7 When, after an overtaking collision of two rods, the last rod to rotate balances on the "direct hit" side of the other rod. its motions to impart its linear motion are contrary and the new linear motion rebounds/reverses perpendicular to the point of contact of the two rods within the imparting rod..


Direct Hits


      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 moving off at separate speeds at different angles from each other, but in the same somewhat general direction..

      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)






>Chapter 3 1P - 2P Collisions