iSoul In the beginning is reality

Speed and its inverse

Physics and transportation have many things in common, in particular they both deal with objects that move or travel. The case of an object that happens to stay in place should be considered as well. An object that may move can be called a vehicle or particle. Let the path, route, or course of an object that may move be called its trajectory.

The speed of an object that may move is defined as the rate of travel distance, that is, the distance traveled per time of travel. The inverse speed, or pace, of an object that may move is defined as the rate of travel time, that is, the travel time per distance traveled.

Note that if you have a non-zero travel distance and a non-zero travel time, then either the speed or the pace may be determined, and the travel distance or the travel time may be in the numerator and or the denominator. However, usually either a travel time or a travel distance is given as an independent variable, with the other as the dependent variable. Then the independent variable should be in the denominator, with the other in the numerator. There are two reasons for this: (1) the independent variable can always be non-zero, whereas the dependent variable might be zero, and (2) the numerator shows what is measured, whereas the denominator shows the units that are used.

For example, 10 meters per second means that in a second of time the object moves 10 meters. A pace of 10 minutes per kilometer means that in a kilometer of space the object takes 10 minutes.

The mean speed is the total distance travelled by all objects in the measurement region, divided by the total time spent in this region. The mean pace is the total travel time by all objects in the measurement region, divided by the total distance travelled in this region. How the mean speed and pace are calculated depends on whether the region is defined spatially or temporally.

The space-mean speed is either the arithmetic mean of the speeds within a spatial region or the harmonic mean of the speeds within a temporal region. The time-mean speed is either the arithmetic mean of the speeds within a temporal region or the harmonic mean of the speeds within a spatial region.

The time-mean pace is either the arithmetic mean of the pace within a temporal region or the harmonic mean of the speeds within a spatial region. The space-mean pace is either the arithmetic mean of the paces within a spatial region or the harmonic mean of the paces within a temporal region.

A basic question is whether or not there is a maximum speed or minimum pace. In transportation there are posted speed limits but whether or not they are obeyed, each vehicle type has a speed beyond which it is physically impossible for it to go. Or, what is equivalent, speed data with values beyond some maximum will be impossible to believe, and so will be ignored or discarded. In physics it has been determined that the speed of light is a maximum speed. So in either case, there is some maximum speed.

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