Where does the radical in the maneuvering speed formula come from?

During primary training, you were likely given a formula to memorize for calculating maneuvering speed when the airplane is loaded at lower than max gross:

$v_{op} = v_a \sqrt{w_{op} / w_{max}}$

where $w_{max}$ is the maximum gross takeoff weight, $v_a$ is the maneuvering speed at max gross, and $w_{op}$ and $v_{op}$ are the operating weight and maneuvering speed at that day’s operating weight.

For small general aviation aircraft, the Pilot’s Operating Handbook or POH may publish only a single maneuvering speed. A key point is this maneuvering speed is valid at max gross only. On checkride day, the Designated Pilot Examiner or DPE will expect the candidate to calculate weight and balance to show that the center of gravity is within a safe range and to calculate maneuvering speed for the combined weight of the airplane, candidate, DPE, fuel, and any baggage on board. When $w_{op}$ decreases, $v_{op}$ does too. (Possible checkride answer: lower operating weight also means slower stall speed.)

But where does the square root sign (“radical”) come from?

The story begins with the lift equation, which according to Chapter 5 of the Pilot’s Handbook of Aeronautical Knowledge (paid link), is

$L = 0.5 \cdot C_L\cdot \rho\cdot V^2\cdot S$

where $L$ is lift, $V$ is velocity, $C_L$ is the coefficient of lift, $\rho$ is air density, and $S$ is the wing’s surface area. In straight-and-level flight, lift must equal weight, so substituting the day’s operating weight $w_{op}$ for $L$, renaming $V$ to $v_{op}$, and swapping sides in the equality results in

$0.5 \cdot C_L\cdot \rho\cdot v_{op}^2\cdot S = w_{op}$

For the same airplane flying around at max gross, the weight and maneuvering speed are different, so they get different unknowns.

$0.5 \cdot C_L\cdot \rho\cdot v_{a}^2\cdot S = w_{max}$

Dividing the first by the second cancels out constants.

$v_{op}^2 / v_a^2 = w_{op} / w_{max}$

Taking the (positive) square root of both sides is where the radical comes from.

$v_{op} / v_a = \sqrt{w_{op} / w_{max}}$

Multiply both sides by $v_a$ to give the (formerly) rote forumla.

$v_{op} = v_a\sqrt{w_{op} / w_{max}}$

The FAA Weight and Balance Handbook (paid link) in Figure 2-15 gives a few examples that we can check our formula against.

• S/N 1005 thru 1147:
  • 2900lbs - 135 KIAS
  • 2600lbs - 126 KIAS (calculated: 127.8)
  • 2200lbs - 116 KIAS (calculated: 117.6)
• S/N 1148 thru 1877 […]:
  • 3000lbs - 131 KIAS
  • 2600lbs - 122 KIAS (calculated: 122.0)
  • 2300lbs - 114 KIAS (calculated: 114.7)

The figures are within 2% for the top lot and off by a fraction of a percent for the bottom.