Matters relating to length and duration in physics and transportation
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
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
The following is based on A Course in Modern Mathematical Physics: Groups, Hilbert Space and Differential Geometry by Peter Szekeres (Cambridge UP, 2004) starting with Example 2.29 on page 54 and modifying it for a temporo-spatial context. The Galilean group. To find the set of transformations of space and time that preserve the laws of
The following is a temporo-spatial modification of the book Mechanics, Third Edition, Volume I of Course of Theoretical Physics by L. D. Landau and E. M. Lifshitz, (Butterworth-Heinenann, Oxford, 1976). [Page 1] §1. CHAPTER I – THE EQUATIONS OF MOTION §1. Generalised co-ordinates ONE of the fundamental concepts of mechanics is that of a particle¹.
The following derivation of the Lorentz transformation is slightly revised from the Appendix to Henri Poincaré: a decisive contribution to Relativity by Christian Marchal, originally published in French as Henri Poincaré: une contribution décisive à la Relativité in La Jaune et la Rouge, août-septembre 1999. Marchal is the chief engineer of mines at ONERA, the Office
A clock consists of two frames of reference. This is seen in an ordinary analogue clock, which is composed of two parts: The space frame is at rest relative to the observer. The time frame is in uniform angular motion relative to the observer. Measurement of space and time requires both frames. The units marked
