iSoul In the beginning is reality

Speeds and velocities

A common word-problem in arithmetic goes something like this: If someone takes a road trip and for half of the time they go one speed and for the other half they go another speed, how should their average speed be determined? The answer is that the average speed is the arithmetic mean of the two speeds. It is implicitly weighted by the time each speed was driven because the denominator of speed is a unit of time.

In symbols, if the first speed is s1 and the second speed is s2, then the average speed is (s1 + s2) / 2. I call this the mean time-speed, though it is generally known as the time-mean speed since it is averaged in reference to time.

But if someone takes a road trip and for half of the distance they go one speed and for the other half they go another speed, how should their average speed be determined? The answer is that the average speed is the harmonic mean of the two speeds, because then it is weighted by the distance each speed was driven, which is in the numerator.

In symbols, if the first speed is s1 and the second speed is s2, then the average speed is 1/((1/s1 + 1/s2) / 2) = 2 / (1/s1 + 1/s2) = 2 s1 s2 / (s1 + s2). I call this the mean space-speed but it is generally known as the space-mean speed since it is averaged in reference to space.

This becomes clearer if we speak about how slow the vehicle travels rather than how fast, that is, the inverse of speed, which I call the space-speed. This measures the travel time per unit of distance. Since the denominator is now distance, the average of the space-speeds is the arithmetic mean. Inverting this results in an ordinary speed (which I call the time-speed), which equals the harmonic mean of the two speeds in the word problem.

The same procedures apply to angular speed or rotation as well. The arithmetic mean averages two different angular speeds for given time periods. The harmonic mean averages two different angular speeds for given angular distances, i.e., angles or rotations.

What about velocity instead of speed? That is, what happens if the speeds are in different directions? If someone takes a road trip and for half of the time they go one speed East and for the other half they go another speed North, their average speed is the arithmetic mean of the two speeds. But their resultant speed, that is, the magnitude of the resultant vector, is the length of the hypotenuse formed by the triangle with speeds East and North weighted by their duration and divided by the total time. It is in the direction of direct flight from the beginning point to the end point.

If someone takes a road trip and for half of the distance they go one speed East and for the other half they go another speed North, their average speed is the harmonic mean of the two speeds. But their resultant speed, that is, the magnitude of the resultant vector, is the length of the hypotenuse formed by the triangle with speeds East and North weighted by their distance and divided by the total time. It is in the direction of direct flight from the beginning point to the end point.

We can easily picture a map representing the route in space, but it is more difficult to picture a time map representing the route in time. Yet the velocities in space and the velocities in time are vectors which can each be represented on a map, that is, geometrically.

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