Definitions of mass and vass

The conservation of momentum states (see here): For a system of objects, a component of the momentum (p = mv, the mass times the velocity) along a chosen direction is constant, if no net outside force with a component in this chosen direction acts on the system. The corresponding principle for levamentum states: For a system …

Polar plot in space and time

This post builds on others such as this. It’s not unusual to see a map of travel time from a central location, for example, this map of Washington, DC, USA (click to enlarge): Two-dimensions of travel time are represented as in space but could be represented as in time. Here is an example, first of …

Relative velocity and lenticity

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 …

Mathematical methods of classical mechanics, part 2

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 …

Ratios and quotients

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 …

Opposite velocities and lenticities

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 harmonic 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 …

Distance, duration and direction

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. …

Analogue clock analyzed

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 …

Galilean invariance of the wave equation

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 …