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 is considered as well. An object that may move is called a body or particle or vehicle. The path, route, or course of an object in motion is its trajectory.

The speed of a body is defined as the time rate of travel distance, that is, the distance per time of travel. The inverse speed, or pace, of a body is defined as the space rate of travel time, that is, the time per distance of travel.

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 would be in the denominator, with the other in the numerator. There are two reasons for this: (1) the independent variable in a ratio is always 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 metres per second means that in a second of time the object moves 10 metres. A pace of 10 minutes per kilometre means that in a kilometre of distance 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 travel distance in this region. How the mean speed and pace are calculated depends on whether the region is defined by distance or by time.

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 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 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.