Question

You look directly overhead and see a plane exactly 1.45 km above the ground flying faster than the speed of sound. By the time you hear the sonic boom, the plane has traveled a horizontal distance of 2.0 km. See Fig. 12-38. Determine (a) the angle of the shock cone, \(\theta \), and (b) the speed of the plane and its Mach number. Assume the speed of sound is 330 m/s.

Short answer

(a) The angle of shock cone is \(35.94^\circ \).

(b) The speed of plane and Mach number of the plane is \(562.21\;{\rm{m/s}}\), and \(1.70\) respectively.

Step-by-step solution

Determination of angle of shock cone

The angle of the shock cone has described the relation between the speed of sound and the moving system. The sine component of the angle of the shock cone is inverse of the Mach number of the system.

Given information

Given data:

The height of plane from ground is \(h = 1.45\;{\rm{km}}\).

The horizontal distance travelled by the plane is \(d = 2.0\;{\rm{km}}\).

The speed of sound is \(c = 330\;{\rm{m/s}}\).

Evaluation of the angle of the shock cone

Part (a)

The diagrammatical representation of shock cone angle is shown below:

The angle of shock cone can be calculated as:

\(\begin{aligned}{c}\theta &= {\tan ^{ - 1}}\left( {\frac{h}{d}} \right)\\ &= {\tan ^{ - 1}}\left( {\frac{{1.45\;{\rm{km}}}}{{2\;{\rm{km}}}}} \right)\\ &= 35.94^\circ \end{aligned}\)

Hence, the angle of shock cone is \(35.94^\circ \).

Evaluation of the speed and Mach number of the plane

Part (b)

The speed of plane can be calculated as:

\(\begin{aligned}{c}{\rm{sin}}\theta &= \frac{c}{v}\\v &= \frac{c}{{\sin \theta }}\\ &= \frac{{\left( {330\;{{\rm{m}} \mathord{\left/{\vphantom {{\rm{m}} {\rm{s}}}} \right.} {\rm{s}}}} \right)}}{{\sin \left( {35.94^\circ } \right)}}\\ &= 562.21\;{{\rm{m}} \mathord{\left/{\vphantom {{\rm{m}} {\rm{s}}}} \right.} {\rm{s}}}\end{aligned}\)

The Mach number of the plane can be calculated as:

\(\begin{aligned}{c}{\rm{Mach}}\;{\rm{number}}&=\frac{v}{c}\\&= \frac{{562.21\;{{\rm{m}} \mathord{\left/{\vphantom {{\rm{m}} {\rm{s}}}} \right.} {\rm{s}}}}}{{330\;{{\rm{m}} \mathord{\left/{\vphantom {{\rm{m}} {\rm{s}}}} \right.} {\rm{s}}}}}\\ &= 1.70\end{aligned}\)

Hence, the speed of the plane and Mach number of the plane is \(562.21\;{\rm{m/s}}\) and \(1.70\) respectively.