How do I calculate the correct rotation speed for a given density altitude?

Question

The Airlines calculate the rotation speed of every takeoff based off the weight given to them from their operations people and also the density altitude conditions. In General Aviation we don't perform such calculation but usually rely on the Pilot Operating Handbook (POH) recommended rotation speed. However the POH rotation speed is based off of gross weight at sea-level and doesn't take into account high density altitudes. Is there a formula that can predict what the ideal rotation speed will be for a specific density altitude?

Answer 1

The given rotation speed is most often indicated air speed[IAS]. IAS is calibrated airspeed [CAS] plus instrument errors. (Outside of the airspeed guage, which as a stand alone item is usually very accurate, errors can be caused by pitot tube and static port mounting locations and large changes in angle of attack.)

CAS is a measure of dynamic pressure not true air speed. Lift is a direct function of dynamic pressure. So the required CAS for a given amount of lift is independent of the air density. If your stall speed is 50kts CAS at sea level then it will be 50kts CAS at 20,000ft.(However true airspeed will be much higher at 20,000ft.)

Airlines use, in combination with extensive performance charts; weight, air density, and temperature for calculating the true speed and thus the required auto brake settings, engine performance settings, V1 V2 Vx and Vy, and by extension determine the required length of the runway, and obstacle clearance.(All of these are determined by mass, thrust, distance, and time rather than lift alone.) Vr CAS/IAS is primarily dependent on lift.(Weight and flap settings.)

A Small aircraft POH does include charts for determining required distance of roll and 50ft obstacle clearance based on air density and maximum gross weight; many will also include selected gross weights below maximum that you can use for interpolating.

If you wish to adjust the rotation CAS based on weight[lift] the formula is the same as is used for adjusting stall speed. If needed first convert IAS to CAS using the chart in your POH, then use CAS in the following formula.

$$ S_n = S_o * \sqrt{ \frac{W_n}{ W_o} } $$ $\text{ Where $S_o$ is original speed, $S_n$ is new speed, $W_o$ is original weight, and $W_n$ is new weight.}$

In plain text if the mathjax formula doesn't render: Sn = So*sqrt(Wn/Wo) ; speed new, speed original, weight new, weight original

After this use the chart in your POH to convert the CAS to IAS and this is your weight adjusted rotation speed.

However, it is my opinion that reducing Vr is only useful where rolling distance is the only limiting factor, because it may actually harm the distance needed to clear a 50ft obstacle. This is because as soon as you rotate you add induced drag, so acceleration before rotation is better than acceleration after rotation. The question then is about the benefit of climbing earlier at a lower initial angle[below Vx] vs later at the ideal angle[Vx for the current density].

Keep in mind that Vx is both weight and density dependent, but it is not as simple to calculate because it involves the specific aircraft design; drag, engine performance limits, and prop selection being primary variables. Mostly Vx and Vy are roughly calculated and then finalized experimentally. But you need the right instruments to accurately record climb angles and climb rates.(GPS data is simple geometry and cannot compensate for the effect of head wind or air density.)

Vx=Vy at the absolute ceiling[not "service ceiling"] for a given weight, from there Vx decreases and Vy increases as density-altitude decreases. All of the CAS numbers will decrease with decreasing weight but again precisely how much is not simple; but you may be able to interpolate existing data, and if doing experimental test pilot stuff you could extrapolate the established data in small increments.

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I just want to add that transport category aircraft use a lot of automated controls like auto-brakes, auto-throttle, and an FMS which all produce much more consistent and repeatable results than the pilot of a light aircraft could manage. As such for light aircraft the variation between each takeoff roll will be greater than the difference created by doing the extra calculations. And then you have effects of runway slope, runway surface [grooved or smooth, damp, wet, snow, dry], winds, and noise abatement which are all used in the airline's computer.