Question

The shock-wave cone created by a space shuttle at one instant during its re-entry into the atmosphere makes an angle of 58.0° with its direction of motion. The speed of sound at this altitude is 331 m/s. (a) What is the Mach number of the shuttle at this instant, and (b) how fast (in m/s and in mi/h) is it traveling relative to the atmosphere? (c) What would be its Mach number and the angle of its shock-wave cone if it flew at the same speed but at low altitude where the speed of sound is 344 m/s?

Short answer

A)The Mach number of the shuttle is 1.18

B) The speed of the shuttle in m/s is 390 m/s and the speed of the shuttle in mi/h is 872 mi/h. C) The Mach number is 1.13 and the angle of the shock wave is $61.9°$

Step-by-step solution

STEP 1 Formula to Mach number of the shuttle

The formula to Mach number of the shuttle is, $\mathrm{M}=\frac{1}{\sin \mathrm{\alpha }}$where, M is the Mach number of the shuttle.is the angle of the shock wave.

The speed of the shuttle in m/s is $\sin \mathrm{\alpha }=\frac{\mathrm{v}}{{\mathrm{v}}_{\mathrm{s}}}$where, $\mathrm{\alpha }$is the angle of shock wave. v is the speed of sound in the air at that instant, ${\mathrm{v}}_{\mathrm{s}}$is the sonic speed of the shuttle.

The Mach number is$\mathrm{M}=\frac{\mathrm{v}}{{\mathrm{v}}_{\mathrm{s}}}$ where, M is the Mach number, ${\mathrm{v}}_{\mathrm{s}}$is the sonic speed of the shuttle. v is the speed of the sound in air.

Calculate the Mach number of the shuttle

The angle of the shock wave created by the space shuttle is 58.0 and the speed of the sound at that instant is 331 m/s. Substitute 58.0 for$\mathrm{\alpha }$in the above equation to find M.

$$\begin{gathered} \mathrm{M}=\frac{1}{\sin \left(58.0°\right)} \\ =1.179 \\ =1.18 \end{gathered}$$

Therefore, The Mach number of the shuttle is 1.18

Calculate the speed of the shuttle

The angle of the shock wave created by the space shuttle is 58.0, the speed of the sound at that instant is 331 m/s and the Mach number of the shuttle is 1.18.Substitute 58.0 for$\mathrm{\alpha }$ and 331 m/s for v in the above equation to find${\mathrm{v}}_{\mathrm{s}}$

$$\begin{gathered} \sin \left({58.0}^{\circ }\right)=\frac{\mathrm{V}}{{\mathrm{v}}_{\mathrm{s}}} \\ {\mathrm{v}}_{\mathrm{s}}=\frac{331\mathrm{m}/\mathrm{s}}{\sin \left({58.0}^{\circ }\right)} \\ =390.31\mathrm{m}/\mathrm{s} \\ =390\mathrm{m}/\mathrm{s} \end{gathered}$$

Formula, to calculate the speed of the shuttle in mi/h is,

$$\begin{gathered} {\left({\mathrm{v}}_{\mathrm{s}}\right)}_{\mathrm{mijh}}=\left({\mathrm{v}}_{\mathrm{s}}\right)\left(\frac{6.21\times {10}^{-4}\mathrm{mi}}{1\mathrm{m}}\right)\left(\frac{3600\mathrm{s}}{1\mathrm{h}}\right) \\ =(390.31\mathrm{m}/\mathrm{s})\left(\frac{6.21\times {10}^{-4}\mathrm{mi}}{1\mathrm{m}}\right)\left(\frac{3600\mathrm{s}}{1\mathrm{h}}\right) \\ =872\mathrm{mi}/\mathrm{h} \end{gathered}$$

Therefore, the speed of the shuttle in m/s is 390 m/s and the speed of the shuttle in mi/h is 872 mi/h.

Calculate the Mach number and angle of the shock

The Mach number is$\mathrm{M}=\frac{\mathrm{v}}{{\mathrm{v}}_{\mathrm{s}}}$Substitute 390 m/s for${\mathrm{v}}_{\mathrm{s}}$ and 344 m/s for v in the above equation to find M

role="math" localid="1668077721573" $$\begin{gathered} =\frac{390\mathrm{m}/\mathrm{s}}{344\mathrm{m}/\mathrm{s}} \\ =1.13 \end{gathered}$$

Therefore, the Mach number is 1.13

Formula to calculate the angle of the shock is$\mathrm{sin\alpha }=\frac{\mathrm{v}}{{\mathrm{v}}_{\mathrm{s}}}$ Substitute 390 m/s for${\mathrm{v}}_{\mathrm{s}}$and 344 m/s for v in the above equation to find$\mathrm{\alpha }$

$$\begin{gathered} =\frac{344\mathrm{m}/\mathrm{s}}{390\mathrm{m}/\mathrm{s}} \\ =61.9° \end{gathered}$$

Therefore, the angle of the shock is$61.9°$