iSoul In the beginning is reality.

# 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 u = Δts. Instantaneous speed is ds/dt. Placepoint pace is dt/ds.

There are two kinds of mean speed or pace: the time mean and the space mean. The time mean is the arithmetic mean if the denominators are a common time interval. The space mean is the arithmetic mean if the denominators are a common space interval. The time mean is the harmonic mean if the denominators are a common space interval. If the denominators are a common time interval, the space mean is the harmonic mean.

The time mean speed (TMS) is the arithmetic mean of speeds with a common time interval. The time mean pace (TMP) is the harmonic mean of paces with a common time interval. For example, the travel distance for vehicles on a highway during a time period is measured. The time mean speed or pace may then be calculated.

The space mean pace (SMP) is the arithmetic mean of paces with a common space interval. The space mean speed (SMS) is the harmonic mean of speeds with a common space interval. For example, the travel time for vehicles over a length of highway is measured. The space mean speed or pace may then be calculated.

If the speed of light is measured by reflection between two points, the result is a time mean speed of the transmission and reflection speeds. For several of these measurements the result is a space mean of time mean speeds. 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 instantaneous 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:

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

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

$SMP(u)=\frac{1}{N}\sum_{i=1}^{N}&space;u_{i}$

$TMP(u)=&space;\left&space;(\frac{1}{N}\sum_{i=1}^{N}&space;u_{i}^{-1}&space;\right&space;)^{-1}$

If a beam of light is reflected back from a known distance, the two-way space 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.