iSoul In the beginning is reality

# Displacement vs. arc length

As pointed out here, average speed does not equal the magnitude of average velocity. But the instantaneous speed does equal the magnitude of instantaneous velocity. For example, the average velocity of one orbit is zero but the average speed is positive.

Consider a section of a curve as below:

The arc length of this section of the curve is Δs. The displacement is Δr. This with the horizontal and vertical differences Δx and Δy makes a triangle. The Pythagorean theorem gives the hypotenuse of the triangle:

(Δr)² = (Δx)² + (Δy)².

Clearly Δs is longer than Δr since the shortest path between two points is a straight line. Now consider a differential version of this:

Then (dr)² = (dx)² + (dy)². But in this case ds is not shorter than dr since they are both differentials, that is, infinitesimals.

So the differential ratios with denominator Δt as Δt → 0 equal dr/dt and ds/dt, which are equal. The instantaneous speed uses either the differential displacement or the differential arc length.

But determination of the arc length is much more difficult than determination of the hypotenuse. So the instantaneous speed, velocity, and acceleration are defined in terms of the hypotenuse, not the arc length. After all, arc length can be determined by integrating the differential hypotenuse.

So the displacement is used instead of the arc length.

# Terminology discussion

In order to describe 3D time some new terms and new meanings for old terms have been introduced in this blog. The reasons for this are discussed in this post.

It would be possible to add a prefix to terms already in use but that over-emphasizes the similarities – or opposition if a negative prefix is used. In some cases, there are existing words that could be easily adopted. Most importantly, there is a need to emphasize that 3D time requires a different way of looking at the world than is commonly done.

# Problems in mechanics, part 3

This post continues a series of problems (part 1 here, part 2 here) based on the website Free Solved Physics Problems, this time concentrating on dynamics problems for time-space corresponding to problems in space-time.

Note: A newton is how much force is required to make a mass of one kilogram accelerate at a rate of one metre per second squared (1 N = 1 kg ⋅ m / s2 ). An oldton is how much rush is required to make a vass of 1 kilogram-1 expedite at a rate of one second per metre squared (1 O = 1 kg-1 ⋅ s / m2).

Problem 6.

A boy of mass 40 kg wishes to play on pivoted seesaw with his dog of mass 15 kg. When the dog sits at 3 m from the pivot, where must the boy sit if the 6.5 m long board is to be balanced horizontally? Solution here.

# Observation and transportation

Impossible objects such as the Necker cube above are drawings that appear as two different objects, in this case either a box standing out toward the lower left or toward the upper right. It can be seen as one or the other but not both simultaneously.

3D space and 3D time are like this. One can see either 3D space or 3D time but not both simultaneously. One may develop a unified 6D geometry for both of them but to measure rates either space or time must be reduced to a scalar or 1D quantity.

It is the same with observation and transportation. One can view a motion from the perspective of an observer (whether one is moving or on the sidelines) or from the perspective of a traveler (whether one is traveling or on the sidelines).

The observer sees motion taking place in 3D space ordered by scalar time. The traveler sees motion taking place in 3D time ordered by scalar space, that is, the stations.

# Time-space introduction

Length measures space and duration measures time. Length is a scalar, which combined with direction describes 3D space. Duration is a scalar, which combined with direction describes 3D time.

In relativity this might be considered trivial; length and duration may be converted into each other by multiplying or dividing by the speed of light. However, the actual measurement will either use rods or clocks, and so be qualitatively different. The act of measurement determines the type of measure more than the units.

Space and time both have three-dimensional geometries. For most purposes one needs to focus on either 3D space or 3D time and ‘scalarize’ the other. Space is scalarized by replacing each point with its distance from a reference point (usually called the origin). Such a scalarized point of 3D space is called a station.

Time is scalarized by replacing each instant with its distime from a reference instant. Such a scalarized instant of 3D time is called a time (as expected).

Space-time means 3D space measured by distance with scalar time. Time-space means 3D time measured by distime with scalar station.

# Formal and material space and time

Science makes no metaphysical claims but it is not unusual for scientists to make metaphysical claims, sometimes even in their scientific publications. That has confused the relationship between science and metaphysics. The philosophies of scientific realism and naturalism have further confused the relationship between science and metaphysics.

As a Christian I must say that if scientists make metaphysical claims, then their metaphysics should be consistent with Christian metaphysics. If scientists object to that, they should refrain from making metaphysical claims.

Science needs to begin without metaphysics. Mathematics has no metaphysics. So science should begin with mathematics. That is, mathematics should be the framework on which science is built.

Thus a science of space and time begins with a mathematical formalism. This formalism should be distinguished from the empirical units employed to measure space and time. As far as I know, that has not been done, so people have confused the measure of length with the form of space and the measure of duration with the form of time.

Isaac Newton separated his metaphysical claims about space and time into what he called scholia in his Principia. He should have just adopted a mathematical formalism and left out any metaphysical claims.

In order to distinguish the formal and material space and time, I have revised the Parallel Glossary for Classical Physics, see link above.

# Space, time, and physical units

Space and time are usually confused with length and duration. That is, the physical units of measure that are typically used with space and time are confused with the pure abstractions of space and time.

Let’s call space and time without physical units abstract space and time. Call space and time with physical units concrete space and time. The distinction is between the use of physical measures, which are after all conventions, and abstractions of space and time, which do not use physical measures. Note that an abstract metric for space or time is also an abstraction, not a physical measure.

There are two kinds of concrete space and time, corresponding to the two kinds of physical units of space and time: length and duration. That is, abstract space may be measured by units of length or duration; abstract time may be measured by units of duration or length.

# Spaces of length and duration

Quantities (called magnitudes) combined with direction are called vectors. Quantities not combined with direction are called scalars. A space is a geometry or topology that contains vectors (which may or may not equal a vector space or Euclidean space as defined in mathematics).

The kind of a space depends on the units of the magnitude. If direction is combined with distance, the result is a distance space, which is 3D space. If direction is combined with duration, the result is a duration space, which is 3D time. Direction may be combined with other quantities, such as speed in a velocity space or pace in a legerity space.

Position vectors are directed from an origin or destination point to a point position. A metric may be defined between positions: distance for distance space and distime for distime space.

# Reduced mass and vass

Here we take the reduced mass and show the parallel reduced vass.

In physics, the reduced mass is the “effective” inertial mass appearing in the two-body problem of Newtonian mechanics. It is a quantity which allows the two-body problem to be solved as if it were a one-body problem.

Given two object bodies, one with mass m1 and the other with mass m2, the equivalent one-body problem, with the position of one body with respect to the other as the unknown, is that of a single body of mass:

${\mu}=\frac{1}{\frac{1}{m_{1}}+\frac{1}{m_{2}}}=\frac{m_{1}m_{2}}{m_{1}+n_{2}}=\frac{1}{n_{1}+n_{2}}$

where the force on this mass is given by the force between the two bodies. This is half of the harmonic mean of the two masses.

# Causes for subjects and objects

This continues posts such as the one here related to Aristotle’s four kinds of cause:

 final cause formal cause efficient cause material cause

A subject is a form with purposes. An object is a material with mechanisms. Objects exist in space-time. Subjects exist in time-space.

The upper causes apply to subjects, who have purposes and plans, destinations and routes. The lower causes apply to objects, which have mechanisms and materials, forces and masses. Though subjects can be considered as objects and objects as subjects.

 why what subjects: final cause formal cause objects: efficient cause material cause

Dynamics is the study of why motion happens, whereas kinematics studies only what motion happens. Kinematics is the material for dynamics. The combination of kinematics and dynamics is called mechanics, but this implies that only objects are considered. If subjects are included, then an alternative term is needed, such as kinedynamics.