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Question 1.11: A commercial airplane flies at 540 mi/h at a standard altitu...

A commercial airplane flies at 540 mi/h at a standard altitude of 30,000 ft. What is its Mach number?

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• Approach: Find the “standard” speed of sound; divide it into the velocity, using proper units.
• Property values: From Table A.6, at 30,000 ft (9144 m), a ≈ 303 m/s. Check this against the standard temperature, estimated from the table to be 229 K. From Eq. (1.38) for air,

a_{ideal  gas} = (kRT)^{1/2}                  (1.38)

a = [kR_{air}T]^{1/2} = [1.4(287)(229)]^{1/2} \approx 303 m/s.

Table A.6 Properties of the Standard Atmosphere
z, m T, K p, Pa ρ, kg/m^3 a, m/s
-500 291.41 107,508 1.2854 342.2
0 288.16 101,350 1.2255 340.3
500 284.91 95,480 1.1677 338.4
1000 281.66 89,889 1.1120 336.5
1500 278.41 84,565 1.0583 334.5
2000 275.16 79,500 1.0067 332.6
2500 271.91 74,684 0.9570 330.6
3000 268.66 70,107 0.9092 328.6
3500 265.41 65,759 0.8633 326.6
4000 262.16 61,633 0.8191 324.6
4500 258.91 57,718 0.7768 322.6
5000 255.66 54,008 0.7361 320.6
5500 252.41 50,493 0.6970 318.5
6000 249.16 47,166 0.6596 316.5
6500 245.91 44,018 0.6237 314.4
7000 242.66 41,043 0.5893 312.3
7500 239.41 38,233 0.5564 310.2
8000 236.16 35,581 0.5250 308.1
8500 232.91 33,080 0.4949 306.0
9000 229.66 30,723 0.4661 303.8
9500 226.41 28,504 0.4387 301.7
10,000 223.16 26,416 0.4125 299.5
10,500 219.91 24,455 0.3875 297.3
11,000 216.66 22,612 0.3637 295.1
11,500 216.66 20,897 0.3361 295.1
12,000 216.66 19,312 0.3106 295.1
12,500 216.66 17,847 0.2870 295.1
13,000 216.66 16,494 0.2652 295.1
13,500 216.66 15,243 0.2451 295.1
14,000 216.66 14,087 0.2265 295.1
14,500 216.66 13,018 0.2094 295.1
15,000 216.66 12,031 0.1935 295.1
15,500 216.66 11,118 0.1788 295.1
16,000 216.66 10,275 0.1652 295.1
16,500 216.66 9496 0.1527 295.1
17,000 216.66 8775 0.1411 295.1
17,500 216.66 8110 0.1304 295.1
18,000 216.66 7495 0.1205 295.1
18,500 216.66 6926 0.1114 295.1
19,000 216.66 6401 0.1029 295.1
19,500 216.66 5915 0.0951 295.1
20,000 216.66 5467 0.0879 295.1
22,000 218.6 4048 0.0645 296.4
24,000 220.6 2972 0.0469 297.8
26,000 222.5 2189 0.0343 299.1
28,000 224.5 1616 0.0251 300.4
30,000 226.5 1197 0.0184 301.7
40,000 250.4 287 0.0040 317.2
50,000 270.7 80 0.0010 329.9
60,000 255.7 22 0.0003 320.6
70,000 219.7 6 0.0001 297.2

• Solution steps: Convert the airplane velocity to m/s:
V = (540 mi/h)[0.44704 m/s/(mi/h)] ≈ 241 m/s.

Then the Mach number is given by
Ma = V/a = (241 m/s)/(303 m/s) = 0.80               Ans.
• Comments: This value, Ma = 0.80, is typical of present-day commercial airliners.

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