Problem 3.15

An observer at sea level does not hear an aircraft that is flying at an altitude of m until it is a distance of km from the observer. Estimate the Mach number at which the aircraft is flying. In arriving at the answer, assume that the average temperature of the air between sea level and 7000 m is ^oC.

Given:
altitude = 7000 m,
distance from observer = 13 km,
\( T_{\text{avg}}= \) -10^oC = 263 K.

To calculate: \( M=? \)

The problem description is schematically shown in Fig. 1.

Fig. 1: Schematic of the problem description

Using trigonometric relation we can estimate the Mach angle as,

\[ \sin\mu=\frac{7000}{13000}=\frac{7}{13} \]

The Mach angle is defined in terms of Mach number as,

\[ \sin\mu=\frac{1}{M}\ \implies\ M=\frac{1}{\sin\mu} \]

\[ M=\frac{13}{7} \]

\[ \boxed{M=1.857}\ . \]

Note that the average temperature is not utilized while arriving to the solution of Mach number, as the calculation of speed of sound is not necessary. Since the average temperature is given, we can estimate the velocity of the aircraft to be,

\[ V=M\,a \]

\[ V=M\,\sqrt{\gamma\,R\,T_{\text{avg}}}=\frac{13}{7}\times\sqrt{1.4\times287\times263} \]

\[ V=603.71\,\text{m/s}\ . \]

Problem 3.15:

Solution: