Consider a particle P in uniform motion. Suppose two inertial observers observe its motion. Observer K is stationary relative to the ground, and observer L is in uniform motion in the same direction as P but at a different rate. (A) Say the spatial position of P relative to K is x(P, K), the spatial …
Part 1 is here. C Measures of motion A motion in RN is a differentiable mapping x: I → RN, where I is an interval on the real axis. The derivative is called the velocity vector at the point r0 ∈ I. The second derivative is called the acceleration vector at the point r0. We …
Mathematical methods of classical mechanics, part 2 Read More »
The traditional ratio, x : y, represents both x / y and y / x. In order to represent a ratio as quotients, both forms are required. Let us define a ratio as an ordered pair of quotients: For vectors this means One can either exclude zero or include infinity as follows; A rate of …
Two opposite velocities — or lenticities — are invariant over time and space. The standard Galileian transformation in the space-time domain is Velocity u transforms as Velocity is not invariant relative to a single inertial observation, but it is relative to observations with opposite relative velocities: That is Harmonic velocities are opposites and so are …
Abstract It is easily shown that there are two kinds of addition for rates: arithmetic addition and reciprocal addition. The kind of addition required depends on whether the variable in common has the same units as the denominator or numerator. This is shown and illustrated with rates of speed and velocity. Several examples are given …
A related post is here. There are three measures of motion: distance, duration, and direction in three dimensions. Direction in three dimensions requires two angles. Distance and duration are non-negative scalars. All measures are relative to an observer. From these base measures several others are derived: Distance divided by duration is a rate called speed. …
This post is related to a previous post here. Consider the dial and one hand of an analogue clock: there are two circular “axes” of reference. One is the circle of the dial, and the other is the circle of the hand (other hands point to the same circle but at different rates): The dial …
This post follows James Rohlf’s Modern Physics from α to Z0 (p.104-105). See also the slides here. The Galilean transformations are applied here to 3D space and 3D time in this case because both space and time are independent arguments. Start with the standard configuration for relativity in which motion is parallel to the x-t axis. The …
This post relates to the previous post Adding and Averaging Rates. A rate is a fraction, though the denominator is often one (a unit rate). In general a rate could be symbolized as Δx/Δy. And so the general addition of rates follows the general addition of fractions: If, as is usual, the denominator is the …
This post is the latest in a series on rates. A rate is a variable quantity measured with respect to a quantity determined independently. A rate is expressed as a ratio of the quantity measured and the independent quantity. A rate of change is a difference of quantities measured with respect to a difference of …
