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I am just wondering how to calculate the Mach number from CAS and Pressure Altitude WITHOUT the use of a flight calculator.

E.g Given FL300 (30000ft) and 325kt CAS find and Mach number.

I have always used an old-school flight computer but now I am just curious about what's the math formulas behind it.

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1 Answer 1

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You can use the following formulas for conversions (source: aerotoolbox.com):

$$ M = \sqrt{5 \left[ \left( \frac{q_c}{P} + 1 \right)^{2/7} - 1 \right]} $$ $$ \text{CAS} = a_0 \sqrt{5 \left[ \left( \frac{q_c}{P_0} + 1 \right)^{2/7} - 1 \right]} $$

with $q_c$ as impact pressure, $ a_0 \approx 340.3 \, \text{m/s} $ as sea level speed of sound and $P_0 \approx 101 \, 325 \, \text{Pa} $ as sea level pressure.

Solving the second equation for $q_c$ gives:

$$ q_c = P_0 \left( \left[ \frac{1}{5} \left( \frac{\text{CAS}}{a_0} \right)^2 + 1 \right]^{7/2} - 1 \right) $$

Plugging your example $\text{CAS} = 325 \, \text{kt} \approx 167.2 \, \text{m/s} $ in gives:

$$ q_c \approx 18 \, 181 \, \text{Pa} $$

At FL300 we have $ P \approx 30 \, 090 \, \text{Pa} $. With the calculated $q_c$ we can now calculate the Mach number with the first equation above:

$$ M \approx 0.8502 $$

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