Bank Angle and the Physics of Standard Rate Turns (continued)

V - Comparing the Approximations to the Exact Formula

At Low Speeds:

Standard Rate Turn Bank Angle (Low Speeds)

Chart 5-1

Chart 5-1 graphs each equation. The solid blue line is the exact equation ^c. At a quick glance it seems that equation remains consistently close to the exact equation throughout the true airspeed (TAS) range. This happens to be the rule of thumb equation outlined in the FAA handbook (see section I).

Standard Rate Turn Bank Angle Error (Low Speeds)

Chart 5-2

In chart 5-2 we can take a closer look at the deviations (errors) from the exact equation . Of the two equations that can be mentally calculated, once again equation is the one with the least amount of error overall throughout the true airspeed range. Equation is more precise than equation up to approximately 100 knots, but then the error increases abruptly after that. Equation is only more precise than the others throughout a very narrow range from about 83 to 100 knots.

At High Speeds:

Standard Rate Turn Bank Angle (High Speeds)

Chart 5-3

From chart 5-3 it seems equation is better over a wider range than the others. It is also evident that equation begins to deviate greatly from the exact equation at close to 600 knots.

Standard Rate Turn Bank Angle Error (High Speeds)

Chart 5-4

Chart 5-4 gives a clearer look at how much each equation is deviating from the exact equation. Even though equation is better over a wider range, equation works better than equation up to about 300 knots. Equation has an error of less than 5^o from 300 knots up until about 580 knots.

Conclusion:

Its is probably best to use equation up to 300 knots. From 300 knots to 580 knots equation is better overall in that range. Above 580 knots it's probably best not to use any of the approximate equations. Beyond around 600 knots the bank angle is going to start to exceed 60^o and it will be highly unlikely that any civilian aircraft is going to want to exceed that bank angle while flying instruments.