Speed Units

Optimizing the performance of Boston Whaler boats
jimh
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Speed Units

Postby jimh » Sat Apr 30, 2022 10:36 am

All speed units use two others units: distance and time. There is little argument about time. The various intervals of time are well known and agreed upon by all users, other than perhaps the unit of day involving one rotation of the Earth. Variations in units of speed are mostly about the unit of distance.

For measuring speed on land, there are two units in common use, statute-miles-per-hour and kilometers-per-hour. For measuring speed through the air or water, the unit knot or nautical-miles-per-hour is often used, in addition to the statute-miles-per-hour or kilometers-per-hour. The three different units of distance:
  • statute mile
  • nautical mile
  • kilometer
The statute mile is based on a statute, an Act of Parliament in England in 1593, which set the mile as 5,280-feet. As for the definition of feet, that unit has an interesting history, but to review it here is not the purpose of this short discussion.

The nautical mile was originally defined as the distance equivalent to a meridional arc length corresponding to one-minute of one-degree of latitude, which required knowing the circumference of the Earth. This is where definitions become interesting.

In creating the length unit of a meter, the fundamental definition was related to the circumference of the Earth. The distance from the North Pole to the Equator measured on a meridian passing through Paris was arbitrarily defined to be 10,000,000-meters. Since the North Pole to Equator distance was one quarter of the Earth circumference, this suggested that the circumference of the Earth would thus be 40,000,000-meters. The actual distance between the North Pole to the Equator on a meridian passing through Paris was never measured. As a substitute, a distance thought to be one-quarter of the meridian from the Equator to the North Pole was measured by a six-year-long survey that concluded in 1798, known as the Arc measurement of Delambre and Méchain. Their survey was much later revealed to have been done with errors which were concealed by a surveyor at the time.

The problem with this method of definition was the Earth was not a perfect sphere. Compensating for an increase in diameter of the sphere at the equator should have been applied.

The full great circle distance of 40,000,000-meters would be 360-degrees. Thus one minute of one degree expressed in meters would be

    40,000,000-meters/(360 × 60) = 1851.851-meters

Later investigation found that the Earth circumference was actually closer to 40,008,000-meters. Thus the length of one-minute of one-degree of Latitude would then be:

    40,008,000-meters/(360 × 60) = 1852.222-meters

The length of a meter was initially defined as a particular length of a platinum bar in Paris, and much later re-defined in terms of atomic wavelengths.

The relationship between the foot and the meter was defined in 1893 using inches as the unit:

    39.37-inches = 1-meters

This defined the foot in terms of meter as

    1-foot = 0.30480061-meter (approximately)


We can calculate the length of a nautical mile in feet from the above relationships. Using the value of one nautical mile as 1852.222-meters:

    1852-meters  × (39.37-inches/1-meter) × (1-foot /12-inches) = 6076.83-feet

However, in the United States and United Kingdom, the circumferential distance of the Earth used to define one-minute of one-degree of Latitude used an average for the ellipsoid distance, resulting a value of 1,853.2480-meters. This produces a value in feet for a nautical mile of approximately:

    1853.2480-meters  × (39.37-inches/1-meter) × (1-foot /12-inches) = 6080.1978-feet

In the United States a five-significant-digit value of 6080.2-feet was used; in the United Kingdom the "Admiralty Mile" is defined as 6,080-feet.

The nautical mile was eventually re-defined to be exactly 1852-meters, more or less losing its previous definition as a particular arc distance length.

Eventually, the United States in 1954 adopted the "international Nautical Mile" value of 1852-meters, converting it into feet with a value using the relationship between the yard and meter then in use, and came up with a value of approximately 6,076.10333-feet in one nautical mile. However, using the international definition of foot, the nautical mile was then adopted as being equivalent to 6,076.11549 International feet.

What I observe in all of this is that the definition of a statute mile has remained the same since 1593, while the definition of the nautical mile, first proposed as a distance of a meridional arc of one-second of one minute has been a moving target when converted to an equivalent length in meters or feet for 450-years.

Limiting use to the United States, the two distance units for speed are generally statute miles (6076.1-feet) and nautical miles (5,280-feet). The conversion factor between these units is thus

    1-nautical mile = (6076.1 / 5280)-statute-miles

    and limiting to four significant digits

    1-nautical mile = 1.151-statute-miles (approximately)

For conversion a value of 1.15 gives quite acceptable accuracy.

In common use the acronym MPH means statute-miles-per-hour and NMPH means nautical-miles-per-hour. The word "knot" as a speed unit is often used ambiguously. While a knot is 1-NMPH, in many cases of casual writing, defining the speed unit as a "knot" leaves confusion about the exact unit being referenced, and, for that reason, explicit use of 1-NMPH is preferred.

The topic of which unit of speed measurement is appropriate for measuring the speed of a small recreational boat capable of speeds of up to 50-MPH will be pursued in the next installment of this article.

jimh
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Joined: Fri Oct 09, 2015 12:25 pm
Location: Michigan, Lower Peninsula
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Re: Speed Units

Postby jimh » Sat Apr 30, 2022 12:59 pm

The concept of using a distance equal to one-minute of one-degree of Latitude for the distance unit "nautical mile" in measurement of speed on the water goes back many centuries. Nautical charts typically were marked with Latitude scales subdivided into minutes. The distance between two positions of the vessel during which it was sailing a straight course line could be easily measured with dividers, then the dividers could be transferred to the Latitude scale on the chart border. The distance would be read by the difference in minutes of Latitude, with 1-nautical mile as the unit. That distance divided by the time it took to sail in hours would be the boat speed in nautical-miles-per-hour.

The difference in time between the two position fixes would provide the time measurement, and the boat speed over ground could be deduced. Boat speed through the water could also be measured with a knot-log device. Comparison between speed-over-ground with speed-through-water would reveal a current effect. These measurements were not dependent on use of any land based distances; land based distance units depend on what country the land was located, and varied from place to place, particularly in the early days of navigation on the open seas.

A vessel making a long voyage on the open seas would measure distances in nautical miles because all the charts used include the Latitude scale that provide an easy method to measure distance in nautical miles from the chart.

There is no argument that, in traditional maritime navigation and continuing today, vessels that make long voyage on the open seas use the nautical mile as the unit of distance.

When operating on inland or near-coastal waters in the United States, many bodies of water explicitly enforce speed limits on vessels in units of statute-miles-per-hour. In Michigan, for example, there is a boat speed limit of 55-MPH.

NATURAL RESOURCES AND ENVIRONMENTAL PROTECTION ACT (EXCERPT)
Act 451 of 1994

324.80146 Maximum or unlimited vessel speed; rules; maximum vessel speed where limits not established; exceptions; resolution requesting reduction in maximum speed limit; emergency conditions; temporary speed limit requirements; time limitations; declaration of state emergency; requiring slow—no wake speed or minimum speed; violation; fine; exceptions; waiver.


(1) The department may promulgate rules to establish maximum motorboat speed limits or to allow unlimited motorboat speed on the waters of this state.

(2) On waters of this state for which a vessel speed limit is not established under subsection (1), for which the department has not established an unlimited vessel speed limit, and for which stricter speed restrictions are not established pursuant to another act, the maximum speed limit is 55 miles per hour, except as follows:
(a) In an emergency as determined by local government authority.
(b) For conservation officers and other peace officers when engaged in official duties.
(c) In the Great Lakes and Lake St. Clair, except for an area within 1 mile of the shoreline measured at a right angle from the shoreline.

Source: https://law.justia.com/codes/michigan/2 ... 80146.html

When operating a boat on inland waters or lakes in which the speed limits imposed are measured in MPH, it hardly make sense to measure boat speed in nautical miles per hour.

SPEED FROM GNSS

It must be understood that a GNSS receiver does not use degrees for its calculations; it uses radians. Also, the GNSS receiver calculates its position based on a space-fixed inertial reference system, which is used to define the satellites' positions. The position of the receiver is calculated in that frame of reference by use of pseudo-ranges from four or more satellites. A concise explanation of how a GNSS receiver operates is given in at https://www.princeton.edu/~alaink/Orf467F07/GNSS.pdf.

There is no assumption of a particular fixed distance for one-minute of one-degree of Latitude in a GNSS receiver. Conversion to an Earth-fixed terrestrial reference is accomplished by use of well-established terrestrial reference systems (such as WGS84). Position can be shown in various Latitude and Longitude angles in degrees, or degrees and minutes, or in degrees and minutes and seconds (which is common in legal documents or surveys), according to the preference of the boat operator.

Most recreational small boats use a GNSS receiver to deduce their position. From distance between position fixes and time between those fixes, speed can be calculated. The speed can be shown in units of MPH, NMPH, or KPH, as the user prefers. On that basis, the use of GNSS navigation does not impose any particular demand that a certain unit of speed must be used. The unit of speed most appropriate for the application should be used.

If making a long-distance open ocean voyage in which there will be no land or national waters nearby, the most appropriate choice of speed unit is still the nautical-mile. If operating in coastal waters where state law applies, the most likely speed unit for the regulatory limits will be statute-miles-per-hour, making MPH the preferred unit for speed. If operating in territorial waters of another nation, the speed unit of that nation may be preferred.

A further consideration in using speed measured by GNSS is the accuracy of the receiver to deduce speed. In most recreational grade GNSS receivers, the speed is deduced from a time interval between two position fixes. The distance between the two positions is computed, and then division by time gives speed. The accuracy of the position fixes determine the accuracy of the distance and the speed calculated. As a general rule, a GNSS receiver with a low Horizontal Dilution of Precision (HDOP), and using an augmentation system to enhance the accuracy of the position fix solutions, the speed calculation will be accurate to about 0.5-MPH in the speed range of typical boats, say 25-MPH. With only one-part-in-50 accuracy of speed, the influence of one-part-in-10,000 in the any conversion factor is not likely to affect the accuracy.