I'm trying to combine two rate of turn formulas, which I found it online this and this.
The first is ROT = 1091 * tan(bank angle) / speed in knots
The second is ROT = speed in knots / radius
By combination,
1091 * tan(bank angle) / speed in knots = speed in knots / radius
speed in knots = [1091 * tan(bank angle) * radius] ^ 0.5
By finding radius, use the simple trigonometry formula c^2 = a^2 + b^2 - 2ab * cos(C)
distance^2 = radius^2 + radius^2 - 2 * radius * radius * cos(bank angle)
distance^2 = 2 * radius^2 * (1 - cos(bank angle))
radius = distance / [2 * (1 - cos(bank angle))] ^ 0.5
Therefore
speed in knots = [1091 * tan(bank angle) * [distance / [2 * (1 - cos(bank angle))] ^ 0.5]] ^ 0.5
Below is the calculation I got. It shows higher bank angle, higher speed and lower turn radius. Does this make sense if the angle-of-attack is fixed? Or I have some wrong calculation? Thanks.
bank angle: 5 10 20 35
distance_in_NM: 2.33 2.33 2.33 2.33
turn_radius: 26.67 13.35 6.70 3.87
speed_in_knots: 50.45 50.67 51.58 54.36
-------------EDIT-------------
The first is still the same. ROT (°/sec) = 1091 * tan(bank angle) / speed in knots
The second is ROT (°/min) = 0.955 * speed in knots / radius
. By converting it to °/sec and thanks by Gerry suggestion, ROT (°/sec) = [0.955 * (speed in knots / 60)] / radius
To combine these two formulas,
1091 * tan(bank angle) / speed in knots = [0.955 * (speed in knots / 60)] / radius
speed in knots = [1091 * tan(bank angle) * radius * 60 / 0.955] ^ 0.5
As a result,
speed in knots = [(1091 * tan(bank angle) * 60 / 0.955) * [distance / [2 * (1 - cos(bank angle))] ^ 0.5]] ^ 0.5
Another attempts with different bank angles. Still higher bank angle, higher speed and lower turn radius.
bank angle: 5 10 20 35
distance_in_NM: 2.33 2.33 2.33 2.33
turn_radius: 26.67 13.35 6.70 3.87
speed_in_knots: 399.91 401.64 408.81 430.89