# Knowing

epistemology, science, kinds of knowledge, methodology

## Opposite velocities and lenticities

Two opposite velocities — or lenticities — are invariant over time and space. The standard Galileian transformation in the space-time domain is Velocity u transforms as Velocity is not invariant relative to a single inertial observation, but it is relative to observations with opposite relative velocities: That is Harmonic velocities are opposites and so are …

Abstract It is easily shown that there are two kinds of addition for rates: arithmetic addition and reciprocal addition. The kind of addition required depends on whether the variable in common has the same units as the denominator or numerator. This is shown and illustrated with rates of speed and velocity. Several examples are given …

## Distance, duration and direction

A related post is here. There are three measures of motion: distance, duration, and direction in three dimensions. Direction in three dimensions requires two angles. Distance and duration are non-negative scalars. All measures are relative to an observer. From these base measures several others are derived: Distance divided by duration is a rate called speed. …

## Analogue clock analyzed

This post is related to a previous post here. Consider the dial and one hand of an analogue clock: there are two circular “axes” of reference. One is the circle of the dial, and the other is the circle of the hand (other hands point to the same circle but at different rates): The dial …

## Galilean invariance of the wave equation

This post follows James Rohlf’s Modern Physics from α to Z0 (p.104-105). See also the slides here. The Galilean transformations are applied here to 3D space and 3D time in this case because both space and time are independent arguments. Start with the standard configuration for relativity in which motion is parallel to the x-t axis. The …

This post relates to the previous post Adding and Averaging Rates. A rate is a fraction, though the denominator is often one (a unit rate). In general a rate could be symbolized as Δx/Δy. And so the general addition of rates follows the general addition of fractions: If, as is usual, the denominator is the …

## Rates and inverses

This post is the latest in a series on rates. A rate is a variable quantity measured with respect to a quantity determined independently. A rate is expressed as a ratio of the quantity measured and the independent quantity. A rate of change is a difference of quantities measured with respect to a difference of …

## An ambiguous problem

Here is a simple word problem: a vehicle travels 80 km in 2 hr, then 60 km in 1 hr. What is its average speed? It is ambiguous because the independent variable is not stated or implied. Was the distance measured based on the time, or was the time measured based on the distance? In …

## Metaphysics of natural science

This is the latest post in a series on science and metaphysics; the previous post is here. The one and only metaphysical postulate of natural science is this: Everything has a fixed nature. This postulate allows the study of classes or kinds or types of things with a common fixed nature. For example, it allows …

## Reciprocal sum of vectors

This post is a slight modification of section 2.0 “The Parallel Sum of Vectors” from W. N. Anderson & G. E. Trapp (1987) “The harmonic and geometric mean of vectors”, Linear and Multilinear Algebra, 22:2, 199-210. We will consider vectors in a real N dimensional inner product space, although some of the results given herein …