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A possibly surprising result is even airplanes as different as a humble Cessna 152 and a Boeing 747, if able to fly the same speed in the same conditions, would use the same pivotal altitude. The airplane’s velocity and acceleration due to gravity are the only factors in determining pivotal altitude. Detailed derivations by John S. Denker and by ERAUSpecialVFR (13:57 YouTube) are included below.

The exact formula for height above the pylon is

$$h = \frac{v_{air}\cdot v_{gnd}}{g}$$

where $v_{air}$ and $v_{gnd}$ are velocities relative to the air and ground, and $g$ is acceleration due to Earth’s gravity. This shows that the common approximation of squaring groundspeed is the calm-day special case.

Assuming we want to know $h$ in feet, we need to connect the building blocks, namely the units of measure, appropriately. Given that $g$ is 32.17405 ft/s² (that is speeding up by 32-ish feet per second every second), compatible velocity will also be denominated in ft/s. To see why, you can think of the units as canceling out, as in

$$\frac{\frac{\textrm{ft}^2}{\textrm{s}^2}\equiv v^2}{\frac{\textrm{ft}}{\textrm{s}^2}\equiv g} \Rightarrow \frac{\textrm{ft}^2}{\textrm{s}^2} \cdot \frac{\textrm{s}^2}{\textrm{ft}} \Rightarrow \textrm{ft}$$

At least in the airplanes I fly, the airspeed indicator displays knots or miles per hour. The conversion factor from knots to feet per second is $\frac{6{,}076}{3{,}600}$ because there are 6,076 feet in a nautical mile and 3,600 seconds per hour. For statue miles to feet per second, the factor is $\frac{5{,}280}{3{,}600}$. Remember that the formula for pivotal altitude has two velocity factors, so we must square the conversion factor.

We are ultimately chasing the denominator, so use the reciprocals of the above conversion factors to get

$$d_{mph} = 32.17405\ \textrm{ft}/\textrm{s}^2 \cdot \Biggl(\frac{3{,}600\ \textrm{s/hr}}{5{,}280\ \textrm{ft/SM}}\Biggr)^2 \approx 14.9569\ \textrm{mph}^2/\textrm{ft}$$

and

$$d_{kts} = 32.17405\ \textrm{ft}/\textrm{s}^2 \cdot \Biggl(\frac{3{,}600\ \textrm{s/hr}}{6{,}076\ \textrm{ft/NM}}\Biggr)^2 \approx 11.2947\ \textrm{knots}^2/\textrm{ft}$$

Take reciprocals to gain clearer insight on what’s happening. In the case of knots, $\frac{1}{11.3}$ is around 0.0885 feet per knots squared. This means approximately the same as 9 feet per knots gained or lost, per 100 knots (because 9 ≈ 0.0885 × 100). Likewise for statute miles, the pivotal altitude changes by about 7 feet per mph airspeed change, per 100 mph the airplane is traveling. In either case, given two airplanes where one is flying twice as fast as another, a unit of airspeed gained for the faster will have twice the impact on its pivotal altitude as compared with its slower counterpart.