This post is related to others, such as here. Consider an analogue clock: The movement of the hand clockwise relative to the dial is equivalent to the movement of the dial couter-clockwise relative to the hand. That is, the motion of the hand relative to the dial corresponds to the opposite motion of the dial […]
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
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
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
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
Independent variables are measured first, independent of other variables. They may be either set to a fixed value or allowed to change at a fixed rate. An example of the former is a race in which the distance is the independent variable set for the race, and of the latter is a time variable, which
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
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 Laws [in the Time Domain] We begin by simply stating in conventional form Newton’s laws of mechanics: I. A
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
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
