Physics of Turns in an Airplane

Below are formulas for calculating airspeed, radius, rate, and bank angle with respect to turns. Simple formulas are presented which relate the items to each other, followed by examples. Each example demonstrates solving for a different unknown. The formulas have a conversion constant, k, which can be adjusted for the units which the reader desires by using the unit analysis following each formula.

  1. Airspeed, Radius of Turn, Rate of Turn
    Airspeed = Radius of Turn * Rate of Turn * k

    k = .01034 or (1/96.7)

  2. Airspeed, Radius of Turn, Angle of Bank

    , where g = gravitation acceleration due to gravity (this is included in k below).

    k = .088537 or (1/11.29)

  3. Rate of Turn, Radius of Turn, Angle of Bank

    By substituting the first formula into the second, we can relate turn rate, radius, and bank angle. However, keep in mind that a particular rate and radius of turn only occur at a particular airspeed. Any problem solved with this formula can be solved with both of the preceding ones.

    tan(Angle of Bank) = Radius of Turn * Rate of Turn^2 * k / g
    (where g = gravitation acceleration due to gravity (this is included in k below))

    k = .000009467 or (1/105,621.01)

  4. Application: Eights on Pylons

    Almost every commercial pilot has been introduced to this formula for determining pivotal altitude in eights-on-pylons:

    Altitude = Groundspeed^2 / 11.3

    This is almost like our formula #2, except what happened to the bank angle? Consider the following:

    An airplane, P, is turning about a point on the ground, G, at bank angle t.

    Any physics handbook will contain the formula for the force of centripetal acceleration (Fa):

    , or solving for R: where W = weight; V = velocity, g = acceleration due to gravity, R = turn radius.

    (The text will have the formula in terms of mass, just remember m = W/g)

    The plane is flying level, so the vertical component of lift equals the weight of the plane. Brushing off trigonometry, we know that Fa, the horizontal component of lift, will equal tan(t) * w. We also know that the the radius R of the circle flown equals the height / tan(t), by the same principle.

    If we set these two equations for R equal, we get:

    By canceling tan(t) and W, we’re left with the following basic formula, that works, regardless of the bank used:

    To use the traditional units of feet and knots, we need to add a conversion constant:

    k = .088537 or (1/11.29)

    That's where the mysterious 11.3 originates.

(C) Copyright 2000, Keith Wannamaker. Please send comments or corrections.

Other helpful references on the subject come from Ed Williams, John Denker, and Ray Preston.