# Mean speed and pace

Speed of a motion is the time rate of length change, that is, the length interval with respect to a timeline interval without regard to direction. Pace of a motion is the space rate of time change, that is, the time interval with respect to a baseline interval without regard to direction.

The symbol for speed is v = Δst and for pace is w = Δts. Instantaneous speed is v = ds/dt. Placepoint (puncstanceous) pace is w = dt/ds.

There are two kinds of mean speed or pace: the arithmetic mean and the harmonic mean. The arithmetic mean is used if the denominators have a common value. The harmonic mean is used if the numerators have a common value.

The arithmetic mean speed (AMS) or simply the mean speed is the arithmetic mean of speeds with a common time interval (i.e., distime). It is also known as the time mean speed. For example, the mean speed for vehicles measured during the same time interval is their arithmetic mean speed.

The harmonic mean speed (HMS) is the harmonic mean of speeds with a common distance. It is also known as the space mean speed. For example, the mean speed for vehicles passing through a common distance is their harmonic mean speed.

Similarly, the arithmetic mean pace (AMP) or simply the mean pace is the arithmetic mean of paces with a common distance. For example, the mean pace for vehicles measured over a common distance is their arithmetic mean pace.

The harmonic mean pace (HMP) is the harmonic mean of paces with a common time interval (distime). For example, the mean pace for vehicles measured during the same time interval is their harmonic mean speed.

If the speed of light is measured by reflection between two points, the result is a harmonic mean speed of the transmission and reflection speeds. There may be speed from many reflections. The second postulate of relativity says that all of these speeds should be equal.

The travel distance for vehicles over a time interval may be measured by probe vehicles that record odometer readings for two points in time. The travel time for vehicles over a length of highway may be measured by a time stamp for each vehicle at two points on the highway. This may be done with sensors placed a short distance apart so the consecutive time stamps are for the same vehicle.

Spot speeds are time speeds of vehicles at a point on the roadway. They may be measured by a radar device on the roadside. The mean speed for spot speeds is a time mean speed.

In symbols:

$AMS(v)=\frac{1}{N}\sum_{i=1}^{N}&space;v_{i}$

$HMS(v)=&space;\left&space;(\frac{1}{N}\sum_{i=1}^{N}&space;v_{i}^{-1}&space;\right&space;)^{-1}$

$AMP(w)=\frac{1}{N}\sum_{i=1}^{N}&space;w_{i}$

$HMP(w)&space;=&space;\left&space;(\frac{1}{N}\sum_{i=1}^{N}&space;w_{i}^{-1}&space;\right&space;)^{-1}$

If a beam of light is reflected back from a known distance, the two-way harmonic mean speed of light may be calculated. From measurements such as this the one-way speed of light in a vacuum has been postulated to be a constant equal to 299,792,458 m/s.