Question

A space probe enters the thin atmosphere of a planet where the speed of sound is only about 42 m/s. (a) What is the probe’s Mach number if its initial speed is 15,000 km/h? (b) What is the angle of the shock wave relative to the direction of motion?

Short answer

(a) The probe’s Mach number is \(99.2\).

(b) The angle of shock wave relative to the direction of motion is \(0.58^\circ \).

Step-by-step solution

Determination of Mach number

The Mach number is described as a fraction of the speed of the moving system to the speed of sound in the medium.

Given data

Given information:

The speed of sound is \(c = 42\;{{\rm{m}} \mathord{\left/{\vphantom {{\rm{m}} {\rm{s}}}} \right.} {\rm{s}}}\).

The initial speed of the probe is \(u = 15,000\;{{{\rm{km}}} \mathord{\left/ {\vphantom {{{\rm{km}}} {\rm{h}}}} \right.} {\rm{h}}}\).

Evaluating the probe’s Mach number

Part (a)

The probe’s Mach number is calculated below:

\(\begin{aligned}{c}{\rm{Mach number}} &= \frac{u}{c}\\ &= \frac{{\left( {15,000\;{{{\rm{km}}} \mathord{\left/{\vphantom {{{\rm{km}}} {\rm{h}}}} \right.} {\rm{h}}}} \right) \times \left( {\frac{{1000\;{\rm{m}}}}{{1\;{\rm{km}}}}} \right) \times \left( {\frac{{1\;{\rm{h}}}}{{3600\;{\rm{s}}}}} \right)}}{{42\;{{\rm{m}} \mathord{\left/{\vphantom {{\rm{m}} {\rm{s}}}} \right.} {\rm{s}}}}}\\ &= 99.2\end{aligned}\)

Hence, the probe’s Mach number is \(99.2\).

Evaluating the angle of shock wave relative to the direction of motion

Part (b)

The angle of shock wave relative to the direction of motion is calculated below:

\(\begin{aligned}{c}\theta &= {\tan ^{ - 1}}\left( {\frac{c}{u}} \right)\\ &= {\tan ^{ - 1}}\left( {\frac{{\left( {42\;{{\rm{m}} \mathord{\left/{\vphantom {{\rm{m}} {\rm{s}}}} \right.} {\rm{s}}}} \right)}}{{\left( {15,000\;{{{\rm{km}}} \mathord{\left/ {\vphantom {{{\rm{km}}} {\rm{h}}}} \right.} {\rm{h}}}} \right) \times \left( {\frac{{1000\;{\rm{m}}}}{{1\;{\rm{km}}}}} \right) \times \left( {\frac{{1\;{\rm{h}}}}{{3600\;{\rm{s}}}}} \right)}}} \right)\\ &= 0.58^\circ \end{aligned}\)

Hence, the angle of shock wave relative to the direction of motion is \(0.58^\circ \).