Question

Consider a large commercial airplane cruising at a speed of 1050 km/h in air at an altitude of 10 km where the standard air temperature is −50°C. Determine if the speed of this airplane is subsonic or supersonic.

Answer 1

Subsonic.

Step-by-step explanation:

Let assume that air behaves ideally. The Mach number is the appropriate criterion to define if speed of the airplane is subsonic or supersonic. This dimensionless number is the ratio of the air plane speed to the sound speed of the fluid. That is to say:

`Ma = (v)/(c)`

Where
`c = \sqrt{(k\cdot R_(u)\cdot T)/(M) }`.

`k` - Specific heat ratio.

`R_(u)` - Ideal gas constant.

`T` - Temperature of atmospheric air.

`M` - Molar mass.

The sound speed of air is:

`c = \sqrt{((1.4)\cdot (8.314\,(kPa\cdot m^(3))/(kmol\cdot K) )\cdot (223.15\,K)\cdot \left((1000\,(m^(2))/(s^(2)) )/(1\,(kJ)/(kg) ) \right))/(28.02\,(kg)/(kmol) ) }`

`c \approx 304.462\,(m)/(s)`

The Mach number is:

`Ma = ((1050\,(km)/(h) )\cdot ((1000\,m)/(1\,km) )\cdot ((1\,h)/(3600\,s) ))/(304.462\,(m)/(s) )`

`Ma = 0.958`

As
`Ma < 1`, the speed of the airplane is subsonic.