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I'm trying to solve for an aircraft's turn radius at mach 3.17 (2432 mph) and a bank angle of 19.8; however, after using various online calculators and personally solving the equation, I keep getting different answers. I'm using the aircraft turn radius equation:

R = V^2/(11.26 ⋅ tan(θ))
R = Aircraft Turn Radius, V = Nm/h,θ = Bank Angle

Plugging in the variables of mach 3.17 and a 19.8 degree bank angle yields the following

R = 2096^2/(11.26 ⋅ tan(19.8)) (mach 3.17 ≈ 2096 nm/h)

R = 4393216/(11.26 ⋅ 0.36002)

R = 4393216/ 4.05384

R = 1,083,717 ft

But after plugging the same variables into various online calculators and getting different results, I'm lost. What did I do wrong?

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    $\begingroup$ Can you link to the calculators that give different results? And show what you entered there? Your calculation seems to be correct. This calculator also gives a very similar result with your numbers. $\endgroup$
    – Bianfable
    Nov 2, 2023 at 12:46
  • $\begingroup$ Hey, thanks for the comment! This is one of the calculators I used, but after plugging in the airspeed (mach 3.17) and bank angle (19.8) it gives a turn radius of "330038.4579509235" feet. Did I just plug these variables into the calculator incorrectly? $\endgroup$
    – Sam Jones
    Nov 2, 2023 at 12:56
  • $\begingroup$ This is how it looks to me. Make sure to select the correct units and it gives ~1,083,715ft. $\endgroup$
    – Bianfable
    Nov 2, 2023 at 13:01
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    $\begingroup$ You say "this", but don't provide a link, or even a name. Makes it hard for others to find. ;) $\endgroup$
    – FreeMan
    Nov 2, 2023 at 13:55
  • $\begingroup$ The calculator hyperlinked in the first comment is one of the calculators I used that gave me a different result. $\endgroup$
    – Sam Jones
    Nov 2, 2023 at 19:27

2 Answers 2

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Units are the root of all evil in most of these situations.

The original equation has a constant of 11.26. This would correspond to a Velocity expressed in some particular unit (knots, mph, km/h, m/s, furlongs/fortnight, fraction of C, whatever) and the output radius in some unit (nm, sm, km, inches, mm, leagues, AU, whatever).

Unless you know that the units of your input match those that the constant applies to, and are understanding the formula's output in the correct units, then it's a case of garbage in - garbage out.

That applies both to your equation and to the online calculators. If a calculator expects its input in mph and delivers its output in yards, but you give it an input in knots & believe the output to be in meters, you'll have a moderately close but definitely incorrect answer.

Unfortunately, not all sources of formulas are as clear about such things as they could be, which is why (since omnipresent calculators render arithmetic errors mostly obsolete) inconsistent units are the most common source of problems such as this.

One related thought: using Mach # for velocity is problematic, because the speed of sound is not a single constant. It varies with temperature, so an aircraft at Mach X.xx at 10,000' has a different speed (expressed as ft/miles/meters/furlongs per second/minute/hour/fortnight) than an aircraft flying at the same Mach number at 30,000' where the temperature is much colder. Entering every calculator with a speed not expressed in Mach will alleviate this issue. Conversions between knots, mph, km/h, m/s, and so on are exact; conversion between any of those and Mach isn't (unless temperature is specified).

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  • $\begingroup$ Comments have been moved to chat; please do not continue the discussion here. Before posting a comment below this one, please review the purposes of comments. Comments that do not request clarification or suggest improvements usually belong as an answer, on Aviation Meta, or in Aviation Chat. Comments continuing discussion may be removed. $\endgroup$
    – DeltaLima
    Nov 3, 2023 at 19:31
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The above answer is fairly accurate but it doesn't address the question. If you're plugging the same numbers into different calculators and getting different answers, the problem is the machine you're using. What is the decimal precision of the device you're using?

The calculator cannot represents every possible number accurately so errors are introduced. That's why when we're expecting to see 1, 0.9999784 sometimes shows up in the display. It becomes very glaring when you're dividing big numbers by small ones. There's a whole field of mathematics that solves this problem, numerical analysis.

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    $\begingroup$ What the questioner is talking about is very big differences, not machine-epsilon differences $\endgroup$
    – sophit
    Nov 3, 2023 at 22:12

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