Instantaneous speed and inverse

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/t. The instantaneous speed is the speed at a point in space […]

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Reciprocal derivative

The reciprocal difference quotient is or The reciprocal derivative of f(x), symbolized by a reversed prime, is the limit of the reciprocal difference quotient as x1 and x2 approach x: or as h approaches zero: The reciprocal derivative of a linear function, f(x) = ax + b, is The reciprocal derivative of a power function,

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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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Harmonic vector realm

This post expands on Harmonic Algebra posted here. A vector space, or better a vector realm, to avoid connecting it with physical space, is a set V on which two operations + and · are defined, called vector addition and scalar multiplication. The operation + (vector addition) must satisfy the following conditions: Closure: If u

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Vector inverse and mean

This post is based on research papers by Anderson and Trapp, Berlinet, and the post on Reciprocal arithmetic. The vector inverse x−1 is defined as with positive norm. For a non-zero scalar k, The reciprocal (or harmonic or parallel) sum is symbolized in various ways, but I prefer a “boxplus” to maintain its relation with addition. The

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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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