Using “Newtonian” space and time, and dividing space by a standard co-ordinate system (the origin being of a totally relative nature).
Quoting from J. C. Maxwell, article 18 of his treatise on matter and motion; “so there is nothing to distinguish one part of space from another except its relation to the place of material bodies… We cannot describe the time of an event except by reference to some other event… or the place of a body except by reference to some other body… All our knowledge, both of time and place is essentially relative”.
This is so for real objects and real observations and/or real experiments, but how about for theoretical considerations. Consider any particle A in space at an instant of time, and with a constant velocity in a straight line. Let the particles position at this instant define the origin of the co-ordinate grid.
Let the direction of motion of the particle define the X-axis, the other axes are then perpendicular to it (and passing through the point of origin).
For a subsequent instant (after some interval of time) the velocity of the particle and its position can be figured from its previous position, so therefore its theoretical absolute velocity and position. [Classically this couldn’t be applied to any real (observed) system because a point is an area of space (or in real system, total system) may also have superimposed motion ad infinitum. But not so here as all motion is relative to a theoretical position in space and space is motionless. But from a theoretical sense it can be considered because you state particle A as having no other motion than its linear velocity. Or it is at absolute rest if it has no motion.]
As seen in general in this hypothesis the use of absolute motion is not so removed from the calculations as it is in classical mechanics because when PP collide their absolute motion (all motion) must be involved to conserve motion.
Because we assume a particle has a constant velocity, and because no forces are considered and (therefore) all motion is transferred through collisions, and when two rods are in contact the motion is transferred instantly (see appendix B), therefore all velocity is always of a constant nature and no true acceleration exists on a primary level.
So, no calculations involving acceleration are used here.
Absence of motion is absolute rest, that is, over any interval of time (duration) a particle does not change its position in space.
Motion is a particle moving with some velocity, that is, over any interval its position in space changes.
If all points of a rod move in a constant linear direction, the rod has a constant linear motion. Because the rod is assumed immutable, all points of the rod always move in unison.
Rotational motion can be defined for the rod as any uniform circular motion around a point on the rod that has a continuously changing point of rotation.
Also there is orbital/circular motion as described in chapter 3. And faux orbital/circular motion as described in chapter 8.
Plus rods can have compounded motion as explained in chapter 8 under vectors.
CHART A-1 All Possible Motions
A. Simple Motions
B. Combined Motions
1. Linear + linear (compounded linear, ad infinitum)
2. Linear (simple or compounded)+ rotational
3. Linear (simple or compounded) + circular (simple or compounded)
4. Circular + circular (simple or compounded)
5 Torque + Linear (simple or compounded)
6. Torque + Rotational
7 Torque + Rotational + Linear (simple or compounded)
Rotational and circular motions would not combine. Nor rotational plus rotational.
For a rod (A) moving through space with some velocity (S) and rotating with some velocity (R) - if the midpoint of the rod could be as a piece of chalk and the plane it travels in space be as a blackboard, it would, over some time interval, scribe a line as S1. However, its real velocity is as S+R which can be figured with standard geometry (given the initial and final configurations) as in figure A-1.
So the apparent value S1 < S+R, but by definition motion must displace space so where is it to be made up? It can be accounted as space covered simultaneously for the allowable combined motions as above. So the “observed” value for displacement is not equal to the actual total displacement.
Section 4 Contrary Motion Vectors
From chapter 2 – Direct Hits. If the motion of the particles is a single vector of motion then it is as stated in chapter 8, but if either or both particles are moving through space with compounded motions (overlapping) then treat each vector component separately in the calculation of the impact results. So they may stop all of each other’s motion, or they may not, depending on the angles of the motion vectors. For example if B has a single vector and A has two vectors as in figure A-2 then at impact B’s motion is stopped but for A only A1 vector is stopped (both A and B undergo torque & rotation). But the A2 vector component continues, and in pulling away frees up some of B’s original motion to continue, and A will slide relative to B as it torques and rotates. End results therefore will vary depending on the speeds involved.
So only impact
within a 90° range gets compounded. When < 90° the force in the two or more vectors can force movement of the
rod without “pulling it apart”, and still cover the distance and direction
involved. >90° it does not, and would “pull apart” the rod. But this
situation never arises from direct impacts from >90° cause the linear motions in the rods to stop, as described
with figure A-2. However in many situations after rotation this can occur. Figuring for this is below.
Vectors after Torque and Rotation
For vectors after rotation causing midpoint to midpoint contact and other cases as per chapter 2, consider this:
Principle 19.2 For cases where divergence of two linear vectors is > 90o the two motions are not compounded but drawn together 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.
Principle 19.4 If two linear motions within a rod (A) are 180o opposite the motion within the rod, the rod is stationary in space until impact from another rod (B). Then, in principle for a midpoint to midpoint collision, all motion in the direction of the (direct) impact is transferred to the impacting rod (B), and the rod hit (A) goes off with the its other linear motion.
Of course the real case of impact in 19.4 is more involved. B torques and rotates, then causes A to re-rotates on it. Then when A reaches its midpoint, at the instant of transfer of motion it will have 3 linear motions, the two 180 degree opposites and the new one at discharge from B. My take on this is as follows.
Principle 19.5 For combining three linear motions, two of which are in 180 degree opposite directions, each old motion "separately" compounds with 1/2 of the new (non-opposite) motion as it where to two new resultants. Those two resultants compound to a further resultant, and in that process the first 3 linear motions are erased and the resultant consists to the two secondary motions as described.
Principle 19.6 following principle 18, in cases of a rod rotating in contrary directions or a rod torqueing in contrary directions, or torqueing and rotating in contrary directions, the rebound is always at a midpoint (perpendicular) from the two opposing motions, and expressed as a resultant linear motion.
For rotational and circular motion the situation is more puzzling as to a possible explanation in that indeed the linear and rotational motion cause 90° divergence ( that is points on the rod that are going forward and backward simultaneously). But indeed spatially in space the motion is expressed again as a type of overlapping motion. That is in rod A’s rotation on B, A’s linear motion is complete and the rotational motion is added on. So a Principle :
Principle 20: For motions to be compounded they must be expressed in movement “displacing space”, (which can be overlapping) completely and separately as to the distance and direction of each of the compounded motions, relative to the initial position of the rod.