physics

Inverse units, inverse algebra

The use of space (stance) as an independent variable and time as a dependent variable leads to inverse ratios. There is pace instead of speed, that is, change in time per unit of length instead of change in length per unit of time. But a faster pace is a smaller number, which is counterintuitive and …

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Lorentz transformation derivation

The first derivation is similar to here. Lorentz transformations for space with time Let unprimed x and t be from inertial frame K and primed x′ and t′ be from inertial frame K′. Since space is assumed to be homogeneous, the transformation must be linear. The most general linear relationship is obtained with four constant …

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Equivalence principle

Inertia is the property of a body that resists changes in its motion. Inertial mass of a body is the ratio of the applied force divided by the body’s acceleration. Gravitational mass is the mass of a body as measured by its gravitational attraction to other bodies. The Equivalence Principle takes several forms. The Newtonian version …

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Galilean transformations derived

This derivation of the Galilean transformations is similar to that of the Lorentz transformations here. Since space and time are assumed to be homogeneous, the transformations must be linear. The most general linear relationship is obtained with four constant coefficients: A, B, C, and D: x′ = Ax − Bt t′ = Ct − Dx …

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Interchangeability of space and time

The extent of a motion is measured in two ways: by its time (duration) and by its space (length). The relation between these two measures is the subject here. Although a definition of uniform motion was given by Archimedes, Galileo was the first to give a complete definition: Equal or uniform motion I understand to …

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Classical dynamics

The following presents the spatio-temporal and temporo-spatial versions of Newton’s laws based on the book Classical Dynamics of Particles and Systems by Thornton and Marion (Fifth Edition, 2008). Start with page 49, section 2.2: 2.2 Newton’s [Spatio-temporal] Laws We begin by simply stating in conventional form Newton’s laws of mechanics: I. A body remains at …

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Ballistics

Ballistic table based on launching from a height and angle with coasting ascent and descent (no drag, no thrust). Note the handy trigonometry identity for range: 2 sin θ cos θ = sin 2θ. This table is in pdf form here. Spatio-temporal Temporo-spatial Initial space angle = θ Initial time angle = φ Initial height …

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Abstract classical mechanics

The following builds on the book Mathematical Aspects of Classical and Celestial Mechanics, 3rd edition, by Vladimir I. Arnold, Valery V. Kozlov, and Anatoly I. Neishtadt (Springer 2006). Basic Principles of Classical Mechanics (cf. Chapter 1) Space and Time The space where the motion takes place is three-dimensional and Euclidean with a fixed orientation. We …

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Temporo-spatial rest

Speed is the length of travel per unit of duration (or time interval). Spatial rest is a speed of zero. That is, there is no change in location per unit of time. A body does not change location (relative to an inertial observer) while time continues. But temporal rest seems different. It cannot be zero …

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