space & time

Matters relating to length and duration in physics and transportation

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 […]

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Additions of rates

Abstract It is easily shown that there are two kinds of addition for rates: arithmetic addition and harmonic 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

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Length, duration and direction

A related post is here. There are three measures of motion: length, duration, and direction in three dimensions. Direction in three dimensions requires two angles or three rectilinear coordinates. Length and duration are non-negative scalars. All measures are relative to an oriented observer. From these base measures several others are derived: Length divided by time

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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

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Addition of rates

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

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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

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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

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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)

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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

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