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Motion vs. movement

The English words motion and movement are similar. They both have to do with “changing position or going from one place to another.” (Collins English Dictionary)

Then what’s the difference? Here are a few ways of putting it:

motion is used to describe physical properties, while movement is used to describe the qualities of motion. Ref.

motion doesn’t always imply a purpose, and movement usually does. Ref.

The difference is very fine. I would say that movement is déplacement d’un lieu à un autre [displacement from one place to another] whereas motion is le fait de ne pas rester immobile [not to stand still]. But usage and context are crucial. Ref.

People may not be consistent about it but for the purposes here they can be distinguished. Motion is the general term in kinetics, the study of motion. It says nothing about the purpose of a motion, or its origin and destination. Something just happens to change place.

However, movement includes some purpose, some origin and destination. A movement is a complete motion, from beginning to end. So movement would be preferred in the arts and social sciences and motion in the natural sciences.

Physics studies motion. Transportation studies movement. They may both speak about something changing position but there is a different perspective.

A movement is an entity, a thing, not just a change as a motion is. A motion can be studied abstractly but a movement is not fully abstract because it is an entity.

A body has its motion and a movement has its figure. A body is flesh-and-blood 3D, with motion only adding a thin 1D time perspective. A movement has 3D animation and life, with a figure only adding a thin 1D space perspective.

Speed vs. velocity

For some background, see here and here.

Velocity is defined as: “The time rate of change of position of a body; it is a vector quantity having direction as well as magnitude.” Speed is defined as: “The time rate of change of position of a body without regard to direction; in other words, the magnitude of the velocity vector.” (McGraw-Hill Dictionary of Physics, 3rd ed.)

However, it’s not that simple. A common example shows the problem:

When something moves in a circular path (at a constant speed …) and returns to its starting point, its average velocity is zero but its average speed is found by dividing the circumference of the circle by the time taken to move around the circle. This is because the average velocity is calculated by only considering the displacement between the starting and the end points while the average speed considers only the total distance traveled. Wikipedia

So the average speed is not the magnitude of the velocity (which is zero in this case) but something else – the travel distance divided by the travel time.

The question is whether the speed over a finite interval should be the magnitude of the displacement divided by the time interval or the arc length divided by the time interval (i.e., the integral of the norm of the velocity function over the time interval). The answer should be the latter, although the former is implied by the common definition of speed.

It is better to define speed as the ratio of the arc length (travel distance) divided by the arc time (travel time). In short, speed is that which is measured by a speedometer.

Measures of motion

This post follows others such as the one here and here. A background document is here.

One-dimensional kinematics is like traveling in a vehicle, and on the dashboard are three instruments: (1) a clock, (2) an odometer, and (3) a speedometer. In principle the speedometer reading can be determined from the other instruments, so let’s focus on a clock and an odometer. The clock measures time, which will be used to measure travel time. The odometer measures distance, or length, which will be used to measure travel distance.

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Observers and travelers

Let us distinguish between observer-receivers and traveler-transmitters. Although observers can travel and travelers can observe, insofar as one is observing, one is not traveling, and insofar as one is traveling, one is not observing. The main difference is this: traveler-transmitters have a destination but observer-receivers do not (or at least not as observers).

Compare the roles of the driver and the passengers in a vehicle: the driver is focused on the road and traveling to the destination, whereas the passengers are looking out the window and observing things in the landscape. These are two different roles.

Observer-receivers of motion naturally compare the motion observed with the elapsed time. But traveler-transmitters have a destination and naturally compare the travel motion with the elapsed distance, which measures progress toward the destination. Because of this, the frame of mind for observer-receivers is 3D space + 1D time, whereas it is 1D space + 3D time for traveler-transmitters.

Observers of the sky naturally think of celestial bodies as appearing when they are observed, as with celestial navigation. That is, they act as though the light observed arrives in their sight instantaneously.

Transmitters of light naturally think of light as reaching its destination as they transmit it, as with visual communication. That is, they act as though the light transmitted arrives at its destination instantaneously.

This is consistent with having two conventions of the one-way speed of light (previously discussed here). To be consistent with the round-trip speed of light equaling the value, c, for all observers, that implies the following:

For observers: received light is instantaneous but transmitted light travels at the speed c/2.

For travelers: transmitted light is instantaneous but received light travels at the speed c/2.

Although relativity theory is the scientific approach, for everyday life the above speeds make things simpler, and are fully legitimate.

From 1D to 3D in two ways

Among the instruments on a vehicle there may be a speedometer, an odometer, a clock, and a compass, which provide scalar (1D) readings of the vehicle’s location. But what is the location of the vehicle in a larger framework? The compass shows two dimensions must exist on a map of this framework, but of what are they dimensions?

The identity of the two dimensions depends on whether the dimensions are associated with the odometer reading (the travel distance) or the clock reading (the travel time). Let’s represent the travel distance by s, the travel time by t, the speed by v, and the travel direction by angle α clockwise from North.

Consider a simple example in which the vehicle is traveling at a constant speed and not changing direction. Then the ratio of the travel distance to the travel time is a constant, which equals the reading on the speedometer: v = s / t.

The vehicle location may be envisioned in two different kinds of maps: (1) In the first kind of map, which is the familiar one, the travel direction is associated with the travel distance. Then the odometer and compass determine the vehicle location, which may be specified by the polar coordinates (s, α) = s. This ordered pair represents a spatial position vector, s. A velocity vector may be constructed from it as v = s / t.

(2) However, we could just as well associate the travel time with the travel direction. So for the second kind of map, the clock and compass determine the vehicle location, which may be specified by the polar coordinates (t, α) = t. This ordered pair represents a temporal position vector, t. A allegrity vector, u, may be constructed from it as u = t / s.

Let’s look at another simple example. Consider a vehicle on a curve that turns for an angle θ at a constant angular velocity of ω with a turning radius of r. The travel distance on the curve is s = = ωt. The travel time is t = /ω = s/ω. In the first case the spatial vector is s = (r cos(ωt), r sin(ωt)). In the second case the temporal vector is t = (r cos(s), r sin(s)), which is found by reparameterizing by the arc length.

Note that in the first kind of map the travel time remains a scalar, which is not associated with any particular position on the spatial map and so is a universal time. Note that in the second kind of map the travel distance remains a scalar, which is not associated with any particular position on the temporal map and so is a universal distance.

The question, “What time is it?” refers to scalar time, which is associated with all points of 3D space. Similarly, one could ask, “what space is it?” referring to the scalar distance, a 1D space, which is associated with all points of 3D time.

Time and space as scalars or vectors

We need to distinguish between scalar (1D) and vector (3D) versions of both time and space. Motion in scalar (1D) time and scalar (1D) space is measured by clocks and odologes, respectively, and apply throughout the associated vector space or vector time (in Newtonian mechanics).

Scalar space is like scalar time. They are proportional if an odologe with constant pace is used. If celestial bodies are used, they may be not quite proportional.

Motion in vector (3D) time and vector (3D) space is measured as points on a curve (trajectory), which may be decomposed into components. The position vector to each point is its distimement or displacement, respectively.

Each instance of vector space is associated with one point of scalar time, and each instance of vector time is associated with one point of scalar space. A value of scalar time is called the time. A value of scalar space may be called the space because it answers the question, ‘which instance of space is it?’

The travel time of a body between two points of vector time, A and B, may be measured with a stopwatch accompanying the body starting simultaneously with A and ending simultaneously with B. The travel distance of a body between two points in vector space, C and D, is measured with a measuring wheel (odometer) accompanying the body starting at location C and ending at location D.

The speed of a body is the travel distance per unit of travel time. The pace of a body is the travel time per unit of travel distance. The velocity and allegrity include the vector travel direction of the body with the ratios given.

Since the travel time or travel distance may not be available to an observer not on the body, the velocity and allegrity may make use of the scalar time or space in the denominator, respectively.

For the velocity one can substitute the vector travel distance per unit of scalar time. The speed uses the magnitude of the vector travel distance per unit of scalar time.

For the allegrity one can substitute the vector of travel time per unit of scalar space. The pace uses the magnitude of the vector travel time per unit of scalar space.

Direction and time

The measurement of the length of a motion follows the course of motion at its own pace. It is a measurement of something passive, and the motion may be past when the measurement takes place.

Cartesian space lacks direction. The independent axes are just coordinates that describe a passive space. The origin is arbitrary and the direction hidden in the coordinates. There are three coordinates, three dimensions to this physical space.

Motion in space is relative to an origin, and so where the motion is coming from. The wind is coming from a certain direction; that is its direction.

Modern natural science excludes teleology. There are no natural goals, no directions. Nature is passive. If there is any goal-seeking, it must come from outside nature.

The measurement of the time of a motion follows the course of motion at its pace. It is a measurement of something active, in motion while the measurement takes place.

A direction is a command and a course. Go West, young man is a course to take and a direction to follow. Trains are distinguished by their destinations. Their direction is indicated by the last stop. The goal and the direction are the same.

Motion in transportation always has a goal, a direction. Motion is physical, but the goal is part of the motion. There’s always a destination.

The destination is some distance away. It takes time to reach the destination. It makes a difference which direction is taken. There are two directions and one distance, which makes three dimensions.

Time in transportation has three dimensions. It is oriented toward the where the vehicle is going, the destination. The train is going toward a certain direction; that is its direction.

Introduction to 3D time with 1D space

Since Newton, time has been the usual and ultimate independent variable for physics. This contrasts with problems in transportation, where time is often optimized. Whether transporting goods across the world, commuters across town, or athletes to the finish line, length is the independent variable against which time is measured and optimized. If length is taken as the independent variable for physics, a mechanics results that is different from Newton’s but equivalent to it. In what follows we explore the basic kinematics and dynamics with length as the independent variable, first in classical, then in relativistic mechanics.

To represent time on a map, one may use isochrones (time contours), as in this map of tsunami travel time in hours:

Or time may be used instead of length, as in this map of European rail travel times in hours:

Note that in this map it is time that is two dimensional, not space. It looks like a distortion of the spatial representation, but it is not a distortion. It is a time map with units of time rather than units of length. The multidimensionality of time will be a main feature of taking distance as the independent variable.

Length could be included as a distance from a particular place, represented by an iso-distance contour. That is, length is reduced to one dimension if time is the dependent variable.

Units such as natural units may be adopted to equalize the units for length and time, i.e., by adopting a constant modal speed dependent on the mode of travel. However, they are still different measurements. Such units allow constant speeds to be represented the same, whether in space or time, which minimizes the seeming distortion of a time map.

As I pointed out here, an independent variable is given to us and so not in our control, and so may seem to flow on independently. A clock is like that, and it gives us a sense that time flows. But a stopwatch starts and stops at our command. Time no longer flows.

Distance can seem to flow if we allow it to. Consider a hop-on, hop-off transit system with a fixed route. It cycles through its various destinations and then repeats the cycle. The distance between two stops may be read from an odometer by subtracting the earlier from the later distance. This is just like finding the time between two points with a clock. Only here it’s the transit system that flows on, accumulating distance indefinitely. Distance flows instead of time.

Posts on space and time chronologically, updated

I previously listed posts on space and time chronologically here. This is a chronological list that includes the posts since then, starting with the most recent (with hyperlinks):

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Distance, duration, and angles

Let’s follow the orbit of a particle or the route of a vehicle as a curvilinear function with associated directions at every point. Measurement produces travel distance r, travel time t, with directions θ and φ. The directions may be considered as functions of either travel distance or travel time: θr, φr, θt, or φt. There are accordingly four possibilities:

(r, t, θr, φr), (r, t, θt, φt), or (t, r, θr, φt), or (t, r, θt, φr).

The latter two may be made equal by a change of convention for measuring the angle. These may be represented rectilinearly as:

(t, rx, ry, rz), (r, tx, ty, tz), (rw, rx, ty, tz), or (tw, tx, ry, rz).

The latter two may be made equal by a change of convention for the axes.

Three possibilities remain: (3D space + 1D time), (1D space + 3D time), or (2D space + 2D time).

An example of the third possibility would be a traveler who measured their horizontal angle relative to magnetic north and their vertical angle relative to the sun. Since magnetic north is (approximately) fixed, it serves to measure the horizontal angle spatially. Since the sun’s position continually changes, it serves to measure the vertical angle temporally. The result is (2+2) with (r, θr) and (t, φt).

Or one could do the opposite and measure the horizontal angle temporally, as with a sundial, and the vertical angle spatially, as with a theodolite. The result is (2+2) with (t, θt) and (r, φr).

If both angles are measured relative to a fixed point, then the result is (3+1) or (t, r, θr, φr). If both angles are measured relative to a moving point, then the result is (r, t, θt, φt). The moving point should be moving at a constant rate, or at least a constant acceleration.

If three coordinates are measured relative to a fixed axis, then the result is (1+3) or (t, rx, ry, rz). If three coordinates are measured relative to a rotating axis, then the result is (r, tx, ty, tz). The moving axis should be moving at a constant rate, or at least a constant acceleration.

The potential reality of (r, t, θr, φr, θt, φt) collapses to one of the possibilities above in the act of measurement. The potential reality of (rx, ry, rz, tx, ty, tz) collapses to one of the rectilinear possibilities above in the act of measurement.