91Ó°ÊÓ

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 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 \(c = 330\;{\rm{m/s}}\).

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 \(35.94^\circ \).

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.