Does the rate of turn increase or decrease by turning at a higher speed?

Question

It is easy to derive a formula for rate of turn in a steady banked turn as:

$$\dot\psi=\frac{g \cdot tan\phi}{v}$$

which indicates that as speed increases rate of turn decreases. This feels counteruntuitive especially given the identity for circular motion:

$$\omega=\frac{v}{r}$$

which would suggest rate of turn increases linearly as speed increases.

Can someone give me some intuition as to why speed has the opposite effect in a banked turn. Given the above I can only think that speed in a banked turn increases radius at a rate greater than it should increase $\omega$ during normal circular motion.

Answer 1

The first formula is a function of bank angle $\phi$ and the velocity $v$. By varying only $v$, you are keeping $\phi$ constant. As a consequence, to keep the turn "steady" and coordinated (that are the assumptions needed to write that formula) you need to increase the radius of turn $r$ and decrease the turn rate.

The second formula is a function of $r$ and $v$. Once again you change only $v$, $r$ will be constant here (while before was $\phi$). This leads to a turn with a higher $\phi$ (again to mantain valid the assumptions behind the formula) and higher rate of turn.

So, to conclude, assuming that you whish to perform a coordinate turn, whether the turn rate increases or decreases is a function of what you whish to keep constant when you change speed:

• if you have a constraint on the bank angle, an increase of the velocity will decrease the turn rate and increase the turn radius
• if you have a constraint on the radius, an increase of the velocity will increase both bank angle and turn rate

To complete the set of equations and see this in practice, you would need to flip each equation and apply it after the other. That means:

• Case 1:

$$\dot\psi=\frac{g \cdot tan\phi}{v} \;\;\;,\;\;\; r = \frac{v}{\dot\psi} \;\;\;,\;\;\; \phi = const.$$

• Case 2:

$$\dot\psi=\frac{v}{r} \;\;\; , \;\;\; \phi = atan\left( \frac{v \cdot \dot\psi}{g} \right) \;\;\; , \;\;\; r= const. $$

Note in case 2 how $\phi$ is computed through an $atan()$ function. This means that increasing $v$ will make $\phi$ approach $90^\circ$. The validity of the model used to derive this equations is no more guaranteed at that point.