The mystery of Vx

Reader Tom Robinson recently wrote me about trying to figure out what the best angle of climb, V_X, is for his 1950 Bellanca. He reported that his flight manual has an “obscure reference” to 80 mph on a takeoff performance chart to clear a 50-foot obstacle but is otherwise glib about this important airspeed. With a published best rate of climb airspeed of 107 mph, a best-angle airspeed of 80 mph to him seemed awfully slow. Tom asked if there is a way to tease out V_X without conducting a sequence of flight tests. I had to smile when I read Tom’s letter as I have read my share of flight manuals that were no less cryptic than his.

Let’s start by considering how these important airspeeds—V_X and V_Y—are determined. If just one value is published, the manufacturer assumes a takeoff power setting, specific aircraft configuration (flaps up or down) and atmospheric conditions. For this discussion we’ll assume that the aircraft flies at maximum gross weight, on a standard day at sea level and with flaps retracted.

For each test case, the aircraft is flown at a range of (calibrated) airspeeds starting from one close to stall speed and continuing in increments of, say, 10 knots to airspeeds beyond cruise but below that which might cause structural problems. The vertical speed is obtained, and the pair (airspeed, vertical speed) is recorded as a data point. Interestingly, the vertical speed indicator is not the best place to look for these data. Instead, the test pilot holds the airplane at an airspeed for a fixed amount of time and the vertical speed is calculated as the altitude gained or lost divided by length of the time interval. After plotting these pairs across the airspeed spectrum, a curve is fitted to the points and becomes the power curve for the conditions flown (see figure 1). Such data are highly influenced by vertical air movement, so the test should be performed in calm conditions and with abundant replication to mitigate their effect. The shape of the power curve is similar for most general aviation aircraft.

The airspeed that creates the maximal vertical speed becomes the published V_Y airspeed to be used whenever the pilot’s goal is to gain altitude efficiently with respect to time. V_X airspeed, on the other hand, maximizes altitude gain with respect to the horizontal distance traveled or, equivalently, the climb rate divided by the forward speed. This latter speed occurs at the airspeed for which a line through the origin and passing through a point on the curve has a maximum slope. So V_X airspeed is necessarily slower than V_Y but faster than stall speed.

I’ve generated a power curve for my own aircraft and getting one that is likely in the ballpark of that generated by the manufacturer took more flights than most owners would care to perform. And even if a very accurate value of V_X can be estimated, the value of V_X appropriate for your situation depends on myriad factors:

Engine performance. The airplane the manufacturer used featured a new engine putting out maximal power. As engines age, the power they produce can wane and this shifts the entire power curve downward. The same phenomenon occurs by flying at high density altitude. V_Y may change slightly—it generally decreases slightly with altitude gain—but V_X will increase toward V_Y (See Figure 2).

Vertical air movement. Rising or falling air will, respectively, shift the polar up or down and will have a similar effect as increased or decreased engine performance. Flying in an updraft means that the pilot can afford to fly at a lower airspeed to clear an obstacle and descending air will require a speed closer to V_Y.

Horizontal air movement. While V_Y remains unaffected by steady, horizontal winds, such is not the case with V_X. Whenever movement is gauged with respect to a fixed object (instead of the air mass itself), ground speed is what matters. When considering the horizontal axis as ground speed, a headwind affords the opportunity to fly at a lower airspeed than the published V_X and a tailwind means that V_X will tend toward V_Y. Taking off into a headwind can greatly increase climb angle, even by sticking with the published V_X airspeed. Unless there is a good reason to depart with a tailwind (for example, a down sloping runway or much lower obstacles such as with a one-way airstrip), departures should always be into the wind.

If you would like to create the power curve for your own aircraft without loads of test flying, author John Denker (see How It Flies, section 7.6.2, av8n.com/how/htm/power.html) offers a way to use a few values from your POH to estimate it. His online text is one of my favorites and offers many other great aviation insights.

If the operating manual doesn’t report a V_X airspeed, as many do not, consulting the takeoff distance chart can indeed shed light on the answer. While the difference between 80 mph V_X and 107 mph V_Y may seem like a lot, it’s not wildly out of line with the 77-knot V_X and 96-knot V_Y published for my Beechcraft Bonanza. It’s also important to see if the performance chart assumes flaps down for a short field effort; if so, that value for V_X would come from a different version of the power curve than the value for (flaps up) V_Y.

In the end, the version of V_X a pilot should use can be mysterious and utterly complicated with changes in engine performance, weight, and atmospheric conditions. While one can afford to use a lower V_X than the book value, slowing down reduces airflow through the engine, degrades forward visibility and decreases the margin over the stall. Taking off into a stiff headwind will do more to increase climb angle than any small deviations in airspeed anyway.

Catherine Cavagnaro teaches aerobatics at UOS and is the Gaston Swindell Bruton Professor of Mathematics at Sewanee: The University of the South.