Mach Number from Temperature

Mach number with temperature correction: Ma = v / v_sound

The Mach number is the ratio of an object's speed to the local speed of sound: Ma = v / v_sound. In air, the speed of sound depends on temperature: v_sound = √(γ·R·T/M) ≈ 331.3 + 0.6·T(°C) m/s. At 20 °C, v_sound ≈ 343 m/s, so Mach 1 ≈ 1,235 km/h. At −50 °C (stratosphere ~11 km, commercial cruise altitude), v_sound ≈ 297 m/s and Mach 1 ≈ 1,070 km/h — that's why high-altitude jets reach the same Mach with much lower true airspeed. Commercial airliners cruise at Mach 0.78–0.85 (transonic regime), the Lockheed SR-71 Blackbird reached Mach 3.2, and the X-15 hit Mach 6.7. Above Mach 1, a shock cone forms, producing the sonic boom; flight above Mach 5 is called hypersonic.

Applications

Aeronautical engineering (transonic and supersonic aircraft design), jet and rocket propulsion, ballistics (supersonic projectiles), atmospheric studies (sound propagation at altitude), and aerodynamic design of missiles and reentry vehicles.

FAQ

Why does Mach 1 change with altitude? Because the speed of sound depends on temperature, not pressure. As you climb into the colder troposphere, v_sound drops, so the same true airspeed corresponds to a higher Mach number.

What is the sonic boom? Above Mach 1 the aircraft compresses air into a shock cone that, when it reaches the ground, is heard as a loud bang — that's why supersonic flight over land is restricted in many countries.

Does humidity affect the speed of sound? Slightly — moist air is less dense than dry air, raising v_sound by less than 1 % at typical conditions. The dominant factor is temperature.