Mach Number Calculator
Enter an object's speed and the local speed of sound to get its Mach number, speed unit conversions, Mach cone angle, and the flow regime (subsonic, transonic, supersonic, hypersonic).
Convert an Object's Speed into a Mach Number
Mach Number Formula
Where:
- \(M\) — Mach number (dimensionless)
- \(v\) — speed of the object relative to the surrounding air (m/s)
- \(a\) — local speed of sound in the surrounding air (m/s)
The formula expresses how many times faster (or slower) than sound an object is travelling through the air. Both speeds must be in the same unit — the calculator converts your inputs to m/s before dividing.
The speed of sound itself depends only on the air temperature:
\[ a = \sqrt{\gamma R T} \]
where \(\gamma = 1.4\) and \(R = 287.058\ \mathrm{J/(kg\cdot K)}\) for air, and \(T\) is the absolute temperature in kelvin.
Mach Number – Calculation Example
Example: an airliner cruises at 900 km/h through air in which sound travels at 343 m/s (roughly sea level at 20 °C).
\[ v = 900\ \mathrm{km/h} = \frac{900}{3.6} = 250\ \mathrm{m/s} \]
\[ M = \frac{v}{a} = \frac{250}{343} \approx 0.729 \]
Result: \(M \approx 0.729\) — the aircraft flies at Mach 0.73, about 73 % of the local speed of sound. This falls in the subsonic regime (0.3 ≤ M < 0.8).
How to Use the Calculator
- Enter the Object speed (v) — the speed of the aircraft, projectile or vehicle relative to the air — and pick its unit (km/h, m/s, mph, ft/s or knots) from the dropdown.
- Enter the Local speed of sound (a) and its unit. If you don't know it, use 343 m/s for warm sea-level air (20 °C), 340 m/s at 15 °C, or about 295 m/s near the stratosphere.
- Read the Mach number, the object speed converted into m/s, km/h, mph and knots, and — for M ≥ 1 — the Mach cone half-angle \(\mu = \arcsin(1/M)\).
- Check the Flow regime classification to see whether the flow is subsonic, transonic, supersonic or hypersonic, along with the implied air temperature for the speed of sound you entered.
What Is the Mach Number?
The Mach number is the ratio of an object's speed to the speed of sound in the surrounding air. It is named after the Austrian physicist Ernst Mach. Because it is a ratio of two speeds, it has no units: Mach 1 means the object moves exactly at the speed of sound, Mach 2 twice as fast, and Mach 0.5 half as fast.
What makes the Mach number so important in aerodynamics is that the speed of sound is not constant. Sound travels at about 343 m/s in warm sea-level air, but only about 295 m/s in the cold stratosphere. Since \(a\) depends only on temperature, the same true airspeed produces a higher Mach number at high altitude. That is why airliners cruising around Mach 0.8 at 11 km altitude fly at roughly 900–950 km/h, while the same Mach number at sea level would require over 1,200 km/h.
The Mach number determines how air behaves around the object. Below about Mach 0.3, air can be treated as incompressible and flow is smooth. As speed rises, compressibility effects appear, and near Mach 1 shock waves form — the source of the sonic boom. This is why engineers classify flight into regimes: subsonic (M < 0.8), transonic (0.8 ≤ M < 1.2), supersonic (1.2 ≤ M < 5) and hypersonic (M ≥ 5). Each regime brings very different drag, heating and structural challenges.
For supersonic flight (M ≥ 1) the calculator also gives the Mach angle \(\mu = \arcsin(1/M)\) — the half-angle of the cone-shaped shock wave trailing the object. The faster the object, the narrower the cone: at Mach 2 the half-angle is 30°, while at Mach 5 it shrinks to about 11.5°.
Reference: Speed of Sound and Flight Regimes
| Air temperature | Speed of sound |
|---|---|
| −55 °C (stratosphere, ~11 km) | ≈ 295 m/s (1,062 km/h) |
| 0 °C | ≈ 331 m/s (1,193 km/h) |
| 15 °C (ISA sea level) | ≈ 340 m/s (1,225 km/h) |
| 20 °C | ≈ 343 m/s (1,235 km/h) |
| 30 °C | ≈ 349 m/s (1,256 km/h) |
Assumptions and Limitations
- The calculator divides two speeds; it assumes you supply the local speed of sound, i.e. the value at the object's altitude and temperature. Using the sea-level value for a high-altitude aircraft understates the true Mach number.
- The speed of the object must be measured relative to the air (true airspeed), not relative to the ground. A tailwind or headwind changes ground speed but not the Mach number.
- The implied temperature in the results assumes dry air with \(\gamma = 1.4\) and \(R = 287.058\ \mathrm{J/(kg\cdot K)}\). Humidity and real-gas effects at hypersonic speeds slightly change the actual speed of sound.
- The regime boundaries (0.3, 0.8, 1.2, 5) are conventional aerospace definitions; the exact transition depends on the object's shape, especially in transonic flow, where local supersonic pockets appear on the wings well before the aircraft itself reaches Mach 1.
If you need the object's average speed from a distance and time, or want to explore other wave phenomena, try our average speed calculator, the Doppler effect calculator, or the wave speed calculator. For high-speed flight you may also find the g-force calculator useful.
Frequently Asked Questions
How do I calculate the Mach number if I only know the speed and air temperature?
First compute the local speed of sound from the temperature: \(a = \sqrt{1.4 \times 287.058 \times T}\) with \(T\) in kelvin (add 273.15 to °C). Then divide the object's speed in m/s by that value. For example, at −50 °C (about 223 K) the speed of sound is roughly 299 m/s, so a 250 m/s aircraft would be at Mach 0.84 — transonic, even though the same speed at sea level is only about Mach 0.73.
How fast is Mach 1, Mach 2 or Mach 0.8 in km/h and mph?
There is no single answer, because Mach numbers are relative to the local speed of sound. At sea level and 15 °C (sound at 340 m/s), Mach 1 ≈ 1,225 km/h (761 mph), Mach 2 ≈ 2,450 km/h (1,522 mph) and Mach 0.8 ≈ 980 km/h (609 mph). At cruising altitude the speed of sound is near 295 m/s, so Mach 0.8 is only about 850 km/h. Always specify altitude and temperature when converting a Mach number to a fixed speed.
What is a typical Mach number for commercial airliners?
Most jet airliners cruise between Mach 0.78 and Mach 0.85 — high subsonic, just below the transonic range. Flying faster would push airflow over the wings supersonic, sharply increasing drag and fuel burn, while flying slower would reduce efficiency. This is why nearly all commercial jets cluster around the same cruising speed regardless of manufacturer.
Why does the same aircraft have a higher Mach number at altitude?
Because the speed of sound falls with temperature, and temperature drops with altitude — down to about −55 °C near 11 km, where sound travels at only ~295 m/s instead of ~343 m/s at sea level. A given true airspeed divided by a smaller speed of sound yields a larger Mach number. That is why pilots monitor Mach number rather than airspeed alone in cruise flight.
Can the Mach number be less than 1 but still cause shock waves?
Yes. In the transonic band (roughly Mach 0.8–1.0) the air accelerates over curved surfaces like wing upper sides, so local flow can exceed Mach 1 even while the aircraft as a whole is still subsonic. This onset is called the critical Mach number, and it produces small shock waves, buffet and a steep drag rise — a key design limit for airliners.
What are the most common mistakes when computing a Mach number?
The biggest is mixing up ground speed with true airspeed — a tailwind changes how fast you move over the ground but not your Mach number. Second is using the sea-level speed of sound (343 m/s) for a high-altitude object, which understates the result. Third is forgetting to convert units: 900 km/h is 250 m/s, not 900 m/s. Make sure both speeds are in the same unit before dividing.
Is Mach number only used for aircraft?
No. It appears wherever an object moves through a compressible gas: rockets and re-entry capsules (hypersonic, above Mach 5), bullets and artillery shells (typically Mach 1–3), jet engine intakes and turbine blades, wind-tunnel testing, and even volcanic ash plumes or meteor shock physics. Note that Mach number applies to gases — the speed of sound in water is about 1,480 m/s, so the same "Mach" speed in water would mean something entirely different.