How to find Radius of Turn given groundspeed and bank angle?

Question

How to find Radius/Rate of Turn given groundspeed and bank angle? Is GS/bank angle one of the formulas? What are the formulas needed for Turns Around a Point? I'm trying to find a way to explain this maneuver as if I was teaching a student pilot?

Answer 1

Here are some equations - (pay attention how one changes the other - to visualize)

Radius of turn When speed doubles, radius increases by 4

$$ \text{Radius of turn} = \frac{v^2}{g \times \tan(\theta)} \qquad \text{(standard units)} $$

Rate of turn

If you increase speed, rate is smaller since you make a bigger turn

$$ \text{Rate of turn} = \frac{g \times \tan(\theta)}{v} $$

Hope this helps

Answer 2

The formula for Turn radius is $V^2 / G_R $, where $V$ is velocity (ground speed), and $G_R$ is radial G, which can be determined by taking the tangent of the bank angle multiplied by what G is in the units you are using for velocity (if velocity is ft/sec, then Gs must be in $\text{ft}/\text{s}^2$ ($\sim 32.2 \text{ft}/\text{s}^2$), or $\sim 9.8 \text{m}/\text{s}^2$). This is to ensure the numbers are all in the same scale.

So turn radius is $V^2 / \tan(\theta)$, where $\theta$ is the Bank Angle.

To get turn rate, just think about the velocity, and the circumference of a complete 360 degree circle. The circumference of a circle with radius $R$ is $2\pi R$, and if the velocity is $V$, then how long will it take the airplane to travel that distance? Time is distance divided by velocity. So it will take Time $T = 2\pi R / V$. And if it takes Time $ 2\pi R / V$ to go 360 degrees, then the turn rate is how many degrees it will change in one second, so that is $360^\circ V / (2\pi R)$.

But $R$ (from turn radius above) is $V^2 / \tan(\theta)$, so turn rate is $ 360^\circ V \tan(\theta) / (2\pi V^2) $, or,

Turn Rate = $ 180^\circ \tan(\theta) / (\pi V) $

NOTE. Again, to use these formulas everything needs to be in the same units: e.g., if V is in $\text{ft}/\text{s}$, then radial G must in $\text{ft}/\text{s}^2$. So in this case, using feet and seconds, you need to add $32 \text{ft}/\text{s}^2$ to the formula for radial G. Instead of $\tan(\theta)$, make it $32\times \tan(\theta)$. If you are entering velocity in knots, then the value of G must be converted to nautical miles per second squared.