Knots, machs, and kilometers per hour — how we measure speed

Speed looks like a simple idea: distance divided by time. And yet humanity learned to measure it in three completely different ways — not on a whim, but because water, solid ground, and the thin air of high altitude obey different laws of physics. At sea we count in knots, on land in kilometers per hour, and high in the atmosphere a pilot abandons both for the dimensionless Mach number. Here is where this trio came from, and why we still live with it.

Sea: a wooden chip and a knotted line

Before GPS and electromagnetic logs, sailors navigated by dead reckoning: knowing the course, the time, and the speed, you could reconstruct your position on the chart. The missing piece was speed — and it was measured with an instrument so simple it was brilliant: the chip log.

The first description of the device was published in 1574 by the English craftsman William Bourne. The log consisted of a wooden chip shaped like a sector of a circle, weighted with lead along its edge. Thrown astern, the chip stood upright, resisted the water, and stayed almost motionless relative to the surface while the ship sailed away, paying out the log line from a reel. The line was divided by literally tied knots — and that is where the name of the unit comes from.

The spacing of the knots was not arbitrary. It followed a simple proportion: the ratio of the distance between knots to the nautical mile had to equal the ratio of the sandglass time to one hour. With the British Admiralty nautical mile (6,080 feet) and a 28-second glass, the spacing worked out to:

x = (6,080 · 28) / 3,600 ≈ 47.3 feet — that is, 47 feet and 3 inches.

A sailor counted the knots slipping through his hand while the sand ran — the count was directly the ship's speed in nautical miles per hour. The method had flaws (swell, currents, the stretch of a hemp line, damp sand running more slowly), and yet it survived for centuries; large sailing ships still used it into the 1920s. Readings were chalked onto a log board and then copied into the log book. That is the origin of all our modern vocabulary for "logging" events and data.

The knot endured because it has an unmatched geometric advantage: a nautical mile is exactly one arcminute of latitude. A ship sailing at one knot covers one arcminute on the chart per hour — navigation and timekeeping fold into one. Since 1929 the unit has been standardized: 1 knot = 1.852 km/h exactly.

Land: the metric revolution and the first speedometers

The kilometer per hour is a direct child of the metric revolution. In 1795 France defined the kilometer, and the meter standard was deposited in the Paris National Archives in 1799. Time, however, was never decimalized — the hour still held 3,600 seconds — so the compound unit km/h emerged naturally in the first half of the 19th century, driven by the railways and the dawn of the automobile.

Speed at first inspired genuine dread. Theories circulated that a passenger racing at more than 20 miles per hour (about 32 km/h) would suffocate from the rush of air. Practice dispelled them: George Stephenson's locomotive Rocket reached about 50 km/h in 1829, and by the end of the century speed had become a measure of progress. In 1898, Count Gaston de Chasseloup-Laubat set the first official land speed record — 63.13 km/h in an electric Jeantaud carriage.

Rising performance forced a way to show the driver the speed. The winner was the eddy-current speedometer, patented in 1902 by the German engineer Otto Schulze. A flexible shaft driven from the gearbox spun a magnet inside an aluminum cup; the rotating field induced eddy currents in the cup, and these deflected a needle in proportion to speed, balanced by a hairspring. The first car with a factory-fitted speedometer was the 1901 Oldsmobile Curved Dash, though the instrument only became standard equipment around 1910.

Speedometers arrived alongside the first speed limits. In interwar Poland the built-up-area limit was usually 40 km/h, but 25 km/h in Warsaw, and just 15 km/h on the narrowest streets; some bridges were restricted to as little as 6 km/h. Outside town, passenger cars long had no limit at all, and the first speed-limit road signs were introduced in Poland only in 1938. The modern standard settled later: the general open-road limit dropped to 90 km/h in 1979, and the urban limit was lowered from 60 to 50 km/h only in the spring of 2004, on Poland's entry into the European Union.

Air: when speed becomes thermodynamics

In flight, both knots and kilometers per hour fail. The basic measuring instrument — the Pitot tube, invented in 1732 by Henri Pitot to gauge the current of the Seine and refined by Henry Darcy in 1858 — does not measure speed directly, but dynamic pressure: the difference between the total (stagnation) pressure at a port facing the oncoming air and the static pressure from side ports.

q = p_total − p_static — dynamic pressure is the difference between total and static pressure.

From this the gauge shows indicated airspeed (IAS). The trouble is that the higher you go, the thinner and colder the air — and then the true airspeed (TAS) rises well above the IAS. Above roughly 300 knots and at high altitude the air stops behaving like an incompressible fluid: pressure builds up ahead of the airframe, corrupting the classic aerodynamic measurement.

That is why high up, pilots switch to the Mach number — named after the 19th-century Austrian physicist Ernst Mach. It is the dimensionless ratio of the true speed v to the local speed of sound a:

Ma = v / a, where a = √(γ · R · T) — the Mach number is the ratio of speed to the local speed of sound.

The key point is that the speed of sound depends only on temperature (γ ≈ 1.4 is the adiabatic index, R = 287.05 J/(kg·K) is the specific gas constant of air, T is temperature in kelvins). Pressure cancels out, because the rise in density at higher pressure is balanced by a rise in stiffness. As a result, at a standard 15 °C at sea level sound travels about 1,225 km/h (340 m/s), but at a jet's cruising ceiling (~11 km, around −57 °C) only 1,062 km/h (295 m/s). Climbing into colder layers at a constant true airspeed, an aircraft increases its Mach number — and approaches the limits of aerodynamics.

That limit is the critical Mach number (Ma_crit): the lowest flight speed at which the air flowing over some part of the airframe (usually the thickest point of the wing) locally reaches the speed of sound. Exceeding it spawns local shock waves, flow separation, and a sharp rise in drag — and, in the extreme, the phenomenon of Mach tuck: the nose pitching down on its own, which the pilot cannot counter with the elevator. That is why fast aircraft have swept wings (which delay the shock waves), and airliners have automatic Mach trimmers that adjust the horizontal stabilizer as the Mach number rises.

It is worth adding that Mach is not a "rigid" unit. The NebulaMath converter uses the conventional value Mach 1 = 343 m/s, the speed of sound in dry air at 20 °C — a convenient "ground-level" reference. In flight the real value of Mach 1 differs at every altitude, because it changes with temperature.

Three units, one table

Although science prefers SI (meters per second), the traditional transport units have proved remarkably durable. The conversions below are exact by definition:

Three relationships are exact equalities: 1 m/s = 3.6 km/h, 1 mph = 1.609344 km/h (since 1959), and 1 knot = 1.852 km/h (since 1929). The rest are derived from these, rounded here to four figures.

Extremes: from creeping continents to a scramjet

The chase for records has always driven engineering, and lining up the extremes shows how much the element decides the outcome. In the thin air of the upper atmosphere you can exceed the speed of sound many times over; at the boundary of water and air, hydrodynamic drag brutally caps performance.

And at the other end of the scale are processes so slow they escape the eye: tectonic plates move on the order of 2–10 cm per year (about 10⁻⁹ m/s), and the Antarctic ice sheet flows toward the ocean at a few meters a year. The same physical quantity — distance over time — spans roughly twelve orders of magnitude: from the drift of continents to a scramjet.

Metrology is a mirror of the element

The knot, the kilometer per hour, and the Mach number are not a matter of national stubbornness or industry tradition. Each of these units grew out of the physics of its environment: the knot ties speed to the geometry of the globe, km/h orders motion over hard, predictable ground, and Mach turns speed into a thermodynamic relation where what matters is not so much the distance covered as the behavior of the airflow around the wing. Today a satellite gives us a reading to a fraction of a km/h — but the three units, shaped by a wooden chip, an eddy-current cup, and a shock wave, still order the way we measure our own motion.

Further reading

• MIT School of Engineering, Why is speed at sea measured in knots? — a concise account of the origin of the knot and the chip log.
• Wikipedia, Chip log and Knot (unit) — the log's construction, line calibration, and the definition of the knot.
• Pilot Institute, Mach Number Explained — the Mach number and its role in aviation, accessibly.
• Wikipedia, Mach tuck and Speed of sound — the physics of the compressibility barrier and the speed of sound in air.
• Wikipedia, List of land speed records — a chronology of land speed records.