Problem 3.17

An observer on the ground finds that an airplane flying horizontally at an altitude of m has traveled km from the overhead position before the sound of the airplane is first heard. Assuming that, overall, the aircraft creates a small disturbance, estimate the speed at which the airplane is flying. The average air temperature between the ground and the altitude at which the airplane is flying is ^oC. Explain the assumptions you have made in arriving at the answer.

Given:
altitude = 2500 m,
distance travelled = 6 km,
\( T_{\text{avg}} = \) 10^oC = 283 K.

To calculate: \( V=? \)

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,

\[ \tan\mu=\frac{\text{altitude}}{\text{distance}}=\frac{2500}{6000}\ \implies\ \mu=\tan^{-1}\left(\frac{2.5}{6}\right) \]

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

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

\[ M=\frac{1}{\sin\left(\tan^{-1}\left(2.5/6\right)\right)} \]

\[ M=2.6\ . \]

The velocity can be calculated as,

\[ V=M\,a \]

\[ V=M\,\sqrt{\gamma\,R\,T_{\text{avg}}}=2.6\times\sqrt{1.4\times287\times283} \]

\[ \boxed{V=876.74\,\text{m/s}}\ . \]

Problem 3.17:

Solution: