Question

Determine the speed of sound in air at 300 K. Also determine the Mach number of an aircraft moving in the air at a velocity of 300 m/s. The gas constant of air is R = 0.287 kJ/kg·K. Its specific heat ratio at room temperature is k = 1.4.

Answer 1

Answer:

$c_3_0_0_K=347.19m/s$

$M=0.864$

Explanation:

The speed of sound in the air increases 0.6 m / s for every 1 ° C increase in temperature.  An approximate speed can be calculated using the following empirical formula:

$c=331.5+0.6\vartheta$

Where:

$\vartheta=T-273.15K$

A more exact equation, usually referred to as adiabatic velocity of sound, is given by the following formula:

$c=\sqrt{k*R*T}$

Where:

$R= Gas\hspace{3}constant\hspace{3}of\hspace{3}air=0.287kJ/kg*K=287J/kg*K\\k=Specific\hspace{3}heat\hspace{3}ratio=1.4\\T=Temperature=300K$

Hence:

$c=\sqrt{(287)*(1.4)*(300)} =347.1887095\approx347.19m/s$

Now, the Mach number at which an aircraft is flying can be calculated by:

$M=\frac{u}{c}$

Where:

$u= Velocity\hspace{3}of\hspace{3}the\hspace{3}moving\hspace{3}aircraft\\c= Speed\hspace{3}of\hspace{3}sound\hspace{3}at\hspace{3}the\hspace{3}given \hspace{3}altitude$

Therefore:

$M=\frac{300}{347.19} =0.8640833984\approx0.864$