Matters relating to length and duration in physics and transportation
An odologe (o′∙do∙loje) is a constant-rate length-measuring device synstancialized with a common waypoint. It is a new coinage from odo(s), path + (horo)loge, clock. In short, it is a clock that shows length instead of time. The simplest odologe takes time from a clock and multiplies it by a conversion speed to produce a length. […]
Many differences are proposed between space and time. This post briefly indicates how all of them are a matter of convention, and so not real. For details, consult posts on this blog. (1) There are three space dimensions but only one time dimension. Directionality can be associated with either length or time (duration). 3D time
J. C. C. McKinsey, A. C. Sugar and P. Suppes (hereafter MSS) wrote “Axiomatic foundations of classical particle mechanics”, (Journal of Rational Mechanics and Analysis, v.2 (1953) p.253-272), which is also described in Suppes’ Introduction to Logic (Van Nostrand, New York, 1957), pp.291-322 (see here). It is only a partial axiomatization of Newtonian mechanics but is
Bodies with space-time orbit by gravitation around their barycenter, the center of mass. The word barycenter is from the Greek βαρύς, heavy + κέντρον, center. The barycenter is one of the foci of the elliptical orbit of each body. For the two-body case let m and M be the two masses, and let r and
For some background, see here and here. Velocity is defined as: “The time rate of change of position of a body; it is a vector quantity having direction as well as magnitude.” Speed is defined as: “The time rate of change of position of a body without regard to direction; in other words, the magnitude
A dimension is informally regarded as the number of coordinates needed to specify the location of a body or point. That may suffice for a mathematical dimension, but a physical dimension is a dimension of something, that is, some unit. In that sense, the dimensions of force are different from the dimensions of velocity. However,
Relativity may be derived as an algebraic relation among differentials. Consider motion in the x spatial dimension, with a differential displacement, dx, differential velocity displacement, dv, and arc (elapsed) time t: dx² = (dx/dt)²dt² = dv²dt² = d(vt)². Let there be a constant, c: dx² = d(vt)² = d(cvt)²/c² = d(ct)² (v/c)² = d(ct)² (1 –
This post follows others such as the one here and here. A background document is here. One-dimensional kinematics is like traveling in a vehicle, and on the dashboard are three instruments: (1) a clock, (2) an odometer, and (3) a speedometer. In principle the speedometer reading can be determined from the other instruments, so let’s
Let the vector r be a displacement vector of motion K that is a parametric function of its arc time t so that r = r(t). Then define s as the arc length of K such that s = s(t) = ∫ || r′(τ) || dτ, where the integral is from 0 to t. Let
This post follows the material on polar coordinates from MIT Open Courseware, here. Instead of the space position vector r, we’ll use the time position vector w, and replace (arc) time with arc length, s. In polar coordinates, the time position of a tempicle A is determined by the value of the radial duration to
