Expert: Paul Soderman - 12/19/2011

Question
I'm a pilot who routinely checks out in different airplanes.  Every Pilots Operating Handbook publishes the gross weight Va and the pilot is expected to reduce that Va based on actual weight.

My question is why do you have to take the square root of the correction ratio of actual wt/gross wt?  Why can't that fraction be mutiplied against Va max without first taking its square root? The velocity at a lower weight will be reduced anyhow without the SQRTF but why is it used?

Answer
David - I am not a pilot and am not familiar with Pilot Operating Handbooks. However, from an aerodynamic standpoint we know that in level flight weight, W, and lift, L, are equal so we can write:

W = L = Cl rho S V^2/2  where Cl = lift coefficient, rho = air density, S = wing area

So for a given aircraft, altitude and lift coefficient, the airspeeds for two different weights are related as:

V2/V1 = SQRT(W2/W1)  where SQRT is square root.

For example, let Vsa = actual stall speed, Vcert = certified stall speed at takeoff gross weight,
Wa =  actual weight, Wcert = takeoff gross weight.  Then:

Vsa/Vcert = SQRT(Wa/Wcert).
Paul