iSoul In the beginning is reality

Measures of speed and velocity

The speed of an object is the ratio of distance (or length) traveled and the duration of travel. It is derived from the distance traveled during a given duration. It is expressed as the measured distance divided by the given duration, that is, distance relative to duration in units of distance over duration, e.g., m/s, km/hr, etc.

For example, the speeds of vehicles passing a fixed point along a roadway may be measured over a given duration by loop detectors and other fixed-location speed detection equipment. These are called spot speeds. The (arithmetic) average of such speeds is called the time mean speed since they are measured during a given period of time. Accordingly, each speed could be called a time speed.

But there is another, complementary way of determining speed. One can select a distance and measure the duration of travel while traversing that distance. Then the measured duration should be in the numerator to show the duration relative to distance, with units s/m, hr/km, etc. Unless the speed is constant, this is not the inverse of the time speed because the distances and durations will not match. It is called the pace, which means the change in time per change in position.

For example, probe vehicles may be in the traffic stream which measure their distance during a set period of time. Or these may be sampled using automatic vehicle location (AVL) data. The harmonic average of such speeds is called the space mean speed since it is measured over a given segment length. Accordingly, each speed could be called a space speed.

Why the harmonic average? Consider each space speed as an inverse speed: put the measured duration of travel in the numerator and the segment length in the denominator, so that the given segment length provides the units for this pace.

Now the average speed may be related to the average pace as follows: invert each speed to put the duration in the numerator and the length in the denominator, take their (arithmetic) average, and invert again to get the average speed. This is the harmonic mean of the space speeds.

Velocity is a vector of speed with the direction of movement. A time velocity may be defined as a velocity whose magnitude is a time speed, and a space velocity as a velocity whose magnitude is a space speed. If its magnitude is a pace, the components are duration divided by length, which is not velocity. It could be called celerity. Why might one use celerity instead of velocity? If the duration is measured for a given length, the celerity gives the appropriate measure: duration relative to length.

What does the direction of the celerity mean? Since it measures duration (relative to a given length), its direction is the temporal direction of movement. This shows again that the same three dimensions may be associated with time (duration) as well as space (length).

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