Introduction

In aircraft performance questions we generally focus on air speed, our airplane's movement with respect to the air mass, but there are reasons for paying attention to our airplane's progress over the earth. During navigation we want to see how we're progressing from Point A to Point B, on the earth. We also care, on a smaller scale, about movement with respect to the ground whenever we deal with two particular V speeds: speed for best (steepest) angle of climb V_x, and speed for best (longest) glide V_bg. With V_x we're trying to clear some approaching earthbound obstacle; with V_bg we're trying to get as far as possible over the ground before contacting it.

While Robert Jordan, in Ernest Hemingway's For Whom the Bell Tolls,reluctantly reported to Pilar, as she questioned him about his making love to Maria, that "Yes, ... It [the earth] moved," and even though geologists report, not at all reluctantly, that the earth's tectonic plates move about an inch a year, we can safely ignore any such movement. So aircraft movement with respect to the earth is due to just two phenomena: (1) aircraft movement with respect to the air; and (2) air movement with respect to the earth. The first gives air speed; the second is wind.

The purpose of this article is to quantify the effect wind has on V_x and V_bg. We'll cover the exact relationship between speeds with respect to the air and with respect to the earth, a couple of approximations to that relation (not limiting for the great majority of general aviation aircraft) and we'll give graphical and formula-based methods for coming up with numerical wind effects. We'll use sample headwinds and tailwinds to demonstrate those effects on a typical Cessna 172. In the end you'll be able to calculate correct values of V_x and V_bg, in reasonable winds, for your particular airplane.

Background

The rule for computing relative motion gets down to this simple vector equation:

Here V stands for velocity, speed together with its direction, stands for the earth, pfor the airplane, and afor the air. The way to remember this rule is to notice the collapse of the "interior" repeated symbol on the right. That's what counts. In English, the rule reads: velocity of the plane with respect to the earth (ground velocity) equals velocity of the air with respect to the earth (wind velocity) plus velocity of the plane with respect to the air (air speed with heading). Of course Eqn. (1) works for any three objects. From the physicist's point of view, there's nothing special about the earth, the air, and airplanes. Aviators know better.

Here's what Eqn. (1) looks like as a vector diagram.

Figure 1 gives less pedantic abbreviations of the some of the velocities and introduces h,"h-dot," as the rate of climb. Symbol is conventionally used for the flight path angle; there and elsewhere parentheses take the places of (unavailable) subscripts to tell us precisely which velocities and angles we're talking about.

There are lots of symbols in Figure 1. Now what are we after? The full-throttle air speed (and, in the gliding case, with a different picture, the no-throttle air speed) V_a (also simply V in the Figure) which, in a given (here) headwind, will give us the largest flight path angle with respect to the earth, _p, henceforth simply _p. It makes sense to call that air speed "V-sub-x in the wind," V_xw. But remember, since we fly solely by reference to air speeds, that V_xw is an air speed. In our graphs and calculations it will be a trueair speed, but it's easy enough to later convert that to more operational calibrated, or even indicated, air speed.

So we want to maximize _p with respect to V. To do that, we're going to need an algebraic relationship between those two variables. One such comes from briefly staring at Figure 1:

We need V_p which, as Figure 1 shows, depends on V_a, V_w and _a. Application of the "law of cosines," the Pythagorean theorem for non-right triangles, gives:

What saves us, in using Eqn. (3), is that _a is a small angle and therefore its cosine is quite close to unity. Even cosine 8, probably a larger climb or glide angle we'll encounter or achieve, only differs from unity by 1%. This small flight path angle approximation is one we've made consistently and is the one largely responsible for the relative simplicity of TBA and of the power-available/power-required (P_a/P_r) analysis. With the cosine set equal to one, the right hand side of Eqn. (3) is a perfect square and we have the simple relationship:

Of course! All we're saying is that, for general aviation aircraft, flight path angles are so shallow --we'll be using the gliding version, too, remember--that there's no practical difference between the (slant) air speed and its horizontal component. Figure 1 doesn't support that statement, but that's because we exaggerated the angles so much. None of these airplanes climbs steadily at 30, not even a Bonanza!

Combining Eqns. (2) and (4), we have

This simple but good approximation will be the basis of the following graphic and analytic calculations of V_xw. In a glide, both h and _p (and its sine) will be negative. But the same relation still holds. It's also true, for small angles, that:

Those two trigonometric functions differ by less than 0.001 up until nearly 8.

The "Exact" Bootstrap Formula for Sin(_p)

To investigate how well Eqn. (5) approximates the "exact" formula, combine Eqns. (2) and (3) to get:

If one uses later Eqn. (8), from The Bootstrap Approach, to evaluate h, our sample Cessna at MSL gives V_x = 56.86 KCAS for a 20 knot headwind, 48.96 KCAS for a 30 knot headwind. Eqn. (5) gives, instead, 56.81 and 48.82 KCAS, respectively. So that approximation is not a problem. We placed 'exact' in quotes, in this sidebar, because TBA uses a small angle approximation itself. Hardly anything is exact in aerodynamics or in airplane performance calculations.

The Graphical Approach

From performance flight tests, or otherwise, assume you have a graph of h, as a function of air speed V, for a given weight and density altitude (or relative air density): h(V;W,). The graph peaks, of course, at V_y (for climbs) or at V_md (for glides). If you start from the origin and lay off various lines hitting the h curve, the one of greatest slope is the one tangent to the curve. If you go steeper, you miss the h curve. See Figure 2. (The somewhat oddly-sized wind speeds are due to our desire to plot calibratedair speeds along the horizontal axis. At 4000 ft density altitude, 20 KCAS = 21.2 KTAS.) Using the approximate "fact" that the slant length from the origin to the curve is the same as the horizontal distance V, Eqn. (5) tells us that the point of tangency is at the speed which is V_x in a calm. Furthermore, starting another tangent line from point (V_w, 0) along the V axis, Eqn. (5) tells us that itspoint of tangency is at the speed for best angle of climb into a headwind of speed V_w, i.e., at V_xw.

So that's all there is to the purely graphical approach to finding V_xw(W,) or V_bgw(W,):

Draw a graph of h(V; W,);

Lay off a tangent from (V_w, 0) to that graph; and

Come straight down from the point of tangency, to the V axis, and read off V_xw (W,) or V_bgw(W,).

Simply treat tailwinds as negative headwinds. If you take starting points for your tangent lines farther and farther to the left, you can see that speed for best angle, V_xw, approaches speed for best rate of climb, V_y.

What about the other way, as V_w gets larger? Eventually our approximation--small path angles --breaks down. As V_w approaches V = V_a, Eqn. (5) blows up. Put into practical terms, in a 60 knot wind you've got bigger problems than worrying about how steep your climb is.

When you're gliding, P_a = 0 and so h = P_r/W and the peak of P_r/W is at V_md. See Figure 3. As you back up the starting point for the tangent, taking larger and larger tailwinds, V_bg approaches V_md. As you take larger headwinds, it looks like V_bg keeps increasing. It does increase, but don't forget that when the wind speed gets to be comparable to the air speed (say about half its size), our approximations break down. When wind speed is higher than your airplane's stall speed you can stay aloft forever.

Speed for best glide V_bg is just the unpowered version of V_x, speed for best angle of climb. Indeed, when gliding, V_bg is your speed for greatest positive (least negative) flight path angle. In similar fashion, V_md is just the unpowered version of V_y. A rate of sink is just a rate of climb which happens to be negative.

The Formula Approach

The Bootstrap Approach furnishes us simple formulas for excess power and for rate of climb:

Here E, K FG, F, G, and H are all "composite" Bootstrap parameters which depend on the nine Bootstrap Data Plate (BDP) parameters and the two variables W and . So once we've tied down the BDP, we're in perfect position to use elementary differential calculus to come up with expressions for V_xw and V_bgw. Solving those expressions will turn out to be something else again.

For reference, Table 1 gives the BDP data used to construct the rate-of-climb graphs of Figures 2 and 3. Gross weight W = 2400 lbf and relative air density (corresponding to h = 4000 ft) = 0.8881. The figure for rated torque M₀ comes from the Lycoming engine's rated power 160 HP at rated speed 2700 RPM. Table 2 gives, for this gross weight and air density, values of the various Bootstrap composite parameters.

Computing the derivative of the right hand side of Eqn. (5), using Eqn. (8), and then setting the resulting expression equal to zero, one gets, for the full-throttle climb,

BDP Item	Symbol	Value	Units
Reference Wing Area	S	174	ft2
Wing Aspect Ratio	A	7.378
Parasite Drag Coefficient	CD0	0.037
Airplane Efficiency Factor	e	0.72
Rated Engine Full-Throttle Torque	M0	311.2	ft-lbf
Loss Proportion Independent of Air Density	C	0.1137
Propeller Diameter	d	6.25	ft
Slope of Linearized Propeller Polar	m	1.70
Intercept of Linearized Propeller Polar	b	0.0564

Table 1. Typical Cessna 172 Bootstrap Data Plate.

In the gliding case, E = F = 0 and so K = G. Then we have the similar equation

Symbol	Value	Units
E	464.7	lbf
F	0.0046508	slug/ft
G	0.0067952	slug/ft
H	1,879,309	ft-lbf2/slug
K	0.011446	slug/ft

Table 2. Composite Bootstrap parameters corresponding to the BDP of Table 1, W = 2400 lbf, relative air density = 0.8881.

While these fifth degree equations can't be solved algebraically, in this form they are easy to solve numerically. Just plug values for V into the right hand sides of either Eqns. (9) or (10) until you get the value of headwind or tailwind desired. That's how we got the solutions given in the captions of Figures 2 and 3. For a more comprehensive look, at three widely-separated altitudes, see Figure 4, where we used Quattro Pro's SolveFor facility to get the numbers.

A common rule of thumb is to adjust calm-wind values of V_x or V_bg by adding or subtracting half the speed of the encountered headwind or tailwind. As the curves (not lines) of Figure 4 show, that rule of thumb greatly overcompensates for light winds; but those are not the problem. But picking on a sizeable 20 knot headwind (and settling on moderate altitude 5000 feet), the graph shows one should back off from calm-wind V_x only about four knots and should add to calm-wind V_bg about six. In a 20 knot tailwind, it suggests adding to calm-wind V_x about two knots and subtracting from calm-wind V_bg about four knots. Except for V_bg and a headwind, the conventional rule of thumb is about twice what's needed. Adjusting by one-fourth of the encountered headwind or tailwind is a better rule.

That revised rule is also supported by calculating the slopes of the graphs at V_w = 0. Evaluating the derivative of V with respect to V_w there (by taking the inverse of the derivative of V_w with respect to V) one finds, in the glide case, that the slope there is exactly 1/4. In the climb case, the derivative is even smaller. But again it's not the very small winds one is concerned about.

Summary

V_x, speed for best (steepest) angle of climb relative to the earth, in comparison with its calm-wind value, is reduced in a headwind and increased in a tailwind. V_bg, speed for best (longest) glide relative to the earth, in comparison with its calm-wind value, is increased in a headwind and reduced in a tailwind. The simple Bootstrap Approach formulas for the aircraft's rate of climb, as a function of air speed, allow one to obtain polynomial functions whose solutions are those V speeds in any given wind condition. But those expressions, of fifth degree, cannot be solved analytically. While numerical solutions are necessary, those are not hard to get.

We presented both graphical and numerical solutions to the relevant equations for a sample Cessna 172 at one weight and at several density altitudes. The corresponding graphs and formulas support a revised

In a headwind (tailwind), reduce V_x (V_bg), relative to its calm-wind value, by one-fourth of the speed of the wind. In a tailwind (headwind), increase V_x (V_bg), relative to its calm-wind value, by one-fourth of the speed of the wind. Only for substantial headwinds (twenty knots or more), double the foregoing prescription for V_bg.