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

VII - Standard Rate with TAS in Kilometers Per Hour (km/hr)

What about if TAS is in kilometers per hour (km/hr)? It is probably not necessary to go in detail over the whole process again, but here are the equations:

As before, we will keep acceleration of gravity (g) in m/s² since it is the most common unit for g. For the remaining units the following table gives each unit and the number that it has to be multiplied by for it to be converted to SI units and used in equation .

Variables Used in Equation in Units Used in Aviation (Speed in km/hr)
	Name Symbol Aviation Unit Multiplier to Convert to SI Exact Conversion      Bank Angle [o] (/180) yes      Rate of Turn [o/s] (/180) yes      Aircraft Speed v [km/hr] (5/18) yes

Table 7-1

Applying all the conversion multipliers in equation we get:

e

More user-friendly names for the symbols yields:

f

Reducing this equation to something more usable. Substituting: the Rate of Turn = 3^o/s for standard rate; use the conventional standard for g, which is 9.80665 m/s²; and evaluating up to 5 significant digits we get:

          Bank Angle = 57.296 · atan (0.0014831 · TAS) ^e

Equation ^e gives the exact (with up to 5 significant digits of precision) bank angle in degrees for a standard rate turn. TAS is the true airspeed in km/h.

Applying our simplification of arctangent, equation ^e becomes:

          Bank Angle 57.296 · 0.0014831 · TAS

Now if we multiply the two numbers in our equation we will have:

          Bank Angle 0.084976 · TAS

Equation does not yield a nice rule of thumb formula with km/hr as it did with knots. Lets try some different formulas and try to figure out an easy way, if there is one, to calculate this estimate formula.

We'll try the following equations that are easy to calculate and are somewhat close to equation :

          0.09 · TAS   

          0.1 · TAS   

          0.1 · TAS -2   

          0.1 · TAS -10   

Comparing the Approximations for Standard Rate to the Exact Formula (in km/hr)

At Low Speeds:

Standard Rate Turn Bank Angle (Low Speeds) (TAS in km/hr)

Chart 7-1

Chart 7-1 graphs each equation. The solid blue line is the exact equation ^e. At a quick glance it seems that the only equations that are easy to calculate and remain close to the exact equation throughout this true airspeed (TAS) range are equations and .

Standard Rate Turn Bank Angle Error (Low Speeds) (TAS in km/hr)

Chart 7-2

In chart 7-2 we can take a closer look at the deviations (errors) from the exact equation. Apart from the exact equation ^e, equation is the one with the least amount of error overall throughout the true airspeed range, but it is not easy to calculate mentally. Equation is more precise than equation by almost two degrees. Ignoring equations and ^e, equation is more precise than the others throughout a range from about 100 to 200 km/hr; then equation becomes more precise until about 320 km/hr; after that equation is more precise but has an error of more than 5^o after around 540 km/hr (see chart 7-4 below).

At High Speeds:

Standard Rate Turn Bank Angle (High Speeds) (TAS in km/hr)

Chart 7-3

From chart 7-3 It looks like the bank angle passes 30^o at around 400 km/hr. The chart below will give a better picture of the errors.

Standard Rate Turn Bank Angle Error (High Speeds) (TAS in km/hr)

Chart 7-4

As mentioned previously, equation is more precise from 320 km/hr but has an error of more than 5^o after around 540 km/hr.

Conclusion:

The best equation will really depend on your speed range:

• From 10 - 100 km/hr equation with a maximum error of 0.5^o.
• From 100 - 200 km/hr equation with a maximum error of 1.5^o.
• From 200 - 320 km/hr equation with a maximum error of 3.5^o.
• From 320 - 540 km/hr equation with a maximum error of 5^o.

To make things even simpler, if you don't mind a little more error from 100 - 200 knots, you can use:

• From 10 - 320 km/hr equation with a maximum error of 3.5^o.
• From 320 - 520 km/hr equation with a maximum error of 5^o.