How Fast is Mach 1?

The speed of sound (Mach 1) at sea level on a 59ºF day is 1,116.4 feet/second or 761.2 MPH or 340.4 meters/second. This is rather pointless to know, as most of us don't fly at sea level; its wet there. For our purposes, though, a rough approximation of the speed of sound is all that's required, and 1,100 ft/sec or 760 MPH or 340 m/sec are close enough. Some of you may know that Mach 1 varies with altitude, temperature and humidity, and a few of those may insist on knowing the exact value of Mach 1 where you are. Ok, you asked for it, but I'm telling you, it ain't worth the effort. Here's the gory details.

The speed of sound is solely determined by the density of the medium though which it travels, and air density changes with its temperature, pressure and humidity. The higher the pressure or lower the temperature or humidity, the denser the air, and the faster the speed of sound. (Unless the altitude is high or the day is really muggy, humidity has a negligible effect on density, so we'll ignore it for the rest of this.) Pressure is inversely related to altitude; the higher you go the less pressure there is. Unfortunately, temperature is also inversely related to altitude; the higher you go the colder it gets. This makes the determination of the speed of sound at a particular altitude a two step process.

First, we determine what the "effective" altitude is based on our current altitude and temperature. From that we can determine the density of the air and therefore the speed of sound. Effective altitudes are measured against standard atmosphere, which at sea level is 59ºF with a density of 0.00237 slugs per cubic foot (15ºC, 1.225 kilograms per cubic meter). As altitude increases, the standard atmostphere gets colder and less compressed. If we assume these standard atmosphere conditions exist at sea level, then at 5,000 feet (1,525 meters) the temperature falls to 41.2ºF (~5ºC) and the density drops to 0.00204 sl/ft3 (1.056 kg/m3), a seemingly significant change.

Standard atmosphere's behavior in this way has been distilled down to a handful of equations that can fairly accurately approximate the temperature and pressure at any given altitude to the edge of space (~45 miles). Airplane pilots need to know this so they can figure out how much lift their wings will generate under the prevailing conditions at their airport.1 This "effective altitude" is actually called the density altitude, becuase its the altitude in standard atmosphere at which that density exists. Once we know the density altitude, we know the air density and can derive the speed of sound.

Use the Javascript-powered calculator below to do just that. Enter an actual altitude and the temperature there in the top row, then press Compute and watch the rest of the numbers come up in the bottom row.

Local Mach 1
Temp
(ºF)
Actual
Altitude (ft)
Density
Altitude (ft)
Density
(sl/ft3)
Speed of
Sound (ft/sec)
OK, now try this. Determine the speed of sound for really dense air, say 40ºF at sea level, and then really thin air, say 90ºF at 7,500 feet. Compare the two. See why we just use an approximation? For most launches its fine, and computing the true speed of sound just isn't worth it. Told ya.

Further Reading

The Atmosphere - A very complete primer on why air acts the way it does.
Air Density and Density Altitude - A great explaination of air density mechanics and why pilots care.
Density Altitude - More on why pilots care about density altitude and what they do about it.

Properties Of The U.S. Standard Atmosphere 1976 - All the details on standard atmosphere

Other Calculators

Density Altitude Calculator - by Design Services
Standard Atmosphere Computations - by Professor Ilan Kroo


1 - A wing's lift generating capacity is also directly related to air density. How much lift it has determines the speed at which the airplane will take off, and thererfore how long its take-off roll must be. As you might imagine, this is a fairly important thing for pilots to know.
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