space & time

Matters relating to length and duration in physics and transportation

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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Two kinds of rates

Rates of change are of two kinds. An ordinary rate for the change of f relative to a unit of x is defined as: The reciprocal rate is the reciprocal of an ordinary rate with a change of g relative to a unit of x is defined as: An ordinary rate has its independent variable …

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Space Time Speed Pace and Mean

Legend: x-axis coordinate, x; length interval, Δx; unit of length, x̂; t-axis coordinate, t; time interval, Δt; unit of time, t̂. For scalars: Rates Speed/Pace Symbols Space Time Ordinary Time speed Δx(t)/t̂ measure length/distance/ displacement given period/unit Alternate Space speed x̂/Δt(x) given length/distance/unit measure interval/period /dischronment Inverse Space pace Δt(x)/x̂ given length/distance/unit measure interval/period /dischronment …

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Instantaneous time and space speed

This post relates to the previous one here. Speed is the time rate of distance traversed. Pace is the space rate of elapsed time. The time speed of a body is the distance traversed per unit of independent time without regard to direction, Δx/t̂. The instantaneous speed is the time speed at a point in …

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Adding and averaging rates

A rate is the quotient of two quantities with different, but related, units. A unit rate is a rate with a unit quantity, usually the denominator. A vector rate is a rate with a vector quantity, usually the numerator. Rates with the same units may be added, subtracted, and averaged. Addition Rates with the same …

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With and between independent variables

This post continues the previous post here on independent and dependent variables. The selection of a physical independent variable (or variables) applies to a context such as an experiment. Within that context all other variables are, at least potentially, dependent on the independent variable(s) selected. Functions with the physical independent variable as a functional independent …

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Distance as an independent variable

A previous post here gives “Motion as a Function of Distance” in which distance is a functional but not a physical independent variable. So distance is an independent variable of the inverse of the function of motion as a function of time. But this functional independence does not change the original independent variable of time. In …

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Independent and dependent variables

There are two kinds of independent variables: (1) functional independent variables, and (2) physical independent variables. To avoid confusion an independent variable it is standard that a variable be of both kinds, since being of one kind does not imply being of the other kind. A physical independent in an experiment remains the independent variable …

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Traffic flow in time and space

The following is based on Wilhelm Leutzbach’s Introduction to the Theory of Traffic Flow (Springer, 1988), which is an extended and totally revised English language version of the German original, 1972, starting with page 3 (with a few minor changes): I.1 Kinematics of a Single Vehicle I.1.1 Time-dependent Description I.1.1.1 Motion as a Function of …

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Dual Galilean transformation

The Galilean transformation is based on the definition of velocity: v = dx/dt, which for constant velocity leads to x = ∫ v dt = x0 + vt So for two observers at constant velocity in relation to each other we have x′ = x + vt with their time coordinates unchanged: t′ = t …

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