# space & time

Explorations of multidimensional space and time with linear and angular motion.

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 …

## Rates of change

The difference quotient is the average rate of change of a function between two points: The instantaneous rate of change is the limit of the difference quotient as t1 and t0 approach each other, which is the derivative of f(t) at that point, denoted by f′(t). Derivatives are added by arithmetic addition, i.e., if f(t) …

## Arithmetic of rates

Euclid wrote that “A ratio is a relation in respect of size between two magnitudes of the same kind.” Magnitudes of different kinds were not considered until Galileo. A ratio between magnitudes a and b is expressed as either a : b or b : a. A quotient is the result of dividing a dividend …

## Two kinds of vector rates

This post builds on the previous one here. Vector rates rates of change are of two kinds. An ordinary rate for the vector change of f relative to a unit of x is defined as: The reciprocal vector rate is the vector reciprocal of an ordinary rate with a vector change of g relative to …