Why Vx and Vy Converge

Dear Rod:

    .... During flight training we have all been taught that Vx increases and Vy decreases with altitude, converging at either the absolute ceiling of the aircraft.  However, in twelve years of flying I have never received a clear explanation of why this is the case. One CFI, to his credit, did some research and then passed on to me a quite technical explanation.  However, since I lacked what appeared to be the requisite Ph.D. in fluid mechanics, my understanding of the subject was, shall we say, only modestly enhanced. Can you offer any help.
 

Thanks,
K.M.
 
 
 

Greetings K.M.:

Your question about Vx and Vy is a good one. In fact, I think this is one of the more challenging aviation topics to explain. Any explanation requires discussing several ideas to provide an adequate answer to the question. So here's my best shot at attempting to explain the concept.

A Detailed Explanation
As you know, Vx is the best angle of climb speed. At this speed the airplane gains the most altitude (vertical movement) for a given distance (horizontal movement) over the ground. Take a look at Figure 1 which depicts an airplane's rate of climb curve for a given true airspeed. (Note: in this article I'll assume that indicated airspeed is the same as calibrated airspeed.)

The blue line is the rate of climb curve. Let's say I placed you in the far left hand corner of the reference box at the 0-FPM and 0-airspeed corner. If I asked you to look up from that position and point to the highest part of the ROC curve that you could see, your arm would point up at the same angle of the tangent line shown in Figure 1. Your arm would point to directly to the tangent point on the graph (identified by the yellow dot). Mathematically speaking, the upward angle of this tangent line represents the maximum amount of vertical gain for a given amount of horizontal movement to the right. But isn't the maximum amount of vertical gain (or altitude gain) for a given distance over the ground also called the best angle of climb? Indeed, it is. So if we drop straight down from the tangent point (yellow dot), we see that the best angle of climb is achieved at a true airspeed of 59 knots at sea level. So Vx is 59 knots true airspeed (TAS) at sea level.

What about the speed for the best rate of climb? Isn't this where you have the greatest amount of altitude gain for a given amount of time. Isn't this just another way of saying that at the best rate of climb speed the VSI needle has the greatest upward deflection? Yes, it is. Therefore, the greatest rate of climb occurs at the very top of the ROC curve (the green dot). If you drop straight down from this point you'll find the speed for the best rate of climb speed (Vy) for this airplane is 76 knots (Figure 1).

ROC Curve Changes Shape With Altitude
Now, ask yourself what happens to the ROC curve when altitude increases. Since the maximum power (thrust) the engine is capable of producing decreases with an increase in altitude, the ROC curve will change shape and grow shorter as shown in Figure 2. The ROC curve for three different altitudes are shown: sea level, 5,000 feet MSL, 10,000 feet MSL. (A nonturbocharged airplane is assumed here.)

It stands to reason that a shorter ROC curve causes the same tangent line to make less of an upward angle with the horizontal. Figuratively speaking, you can think of lowering tangent line as the path of an airplane climbing at less of an angle as a result of an increase in altitude. Therefore, dropping straight down from the tangent point (yellow dot), we find that the best angle of climb speed has increased to 69 knots (TAS) as shown in Figure 3. And this is the basic reason why Vx increases with altitude. (These angles are only symbolic representations. The airplane's climb angle isn't the same upward angle made by the tangent line. This is, nevertheless, a reasonable way to compare ROC curves for the same airplane at two different altitudes.)

In Figure 3, it's also apparent that Vy has increased to 80 knots (drop straight down from the green dot). How can this be? Weren't you told that Vy decreases with an increase in altitude? Yes, but that's only if you're talking indicated airspeed, not true airspeed (I'll explain this shortly). Right now, as far as true airspeed is concerned, Vy increases with an increase in altitude. It increases for several reasons. To visualize why it increases, look at how the ROC curve (as measured by test pilots) changes shape with altitude. The curve shrinks vertically and its top moves to the right slightly (the shape of the ROC curve changes because of changes in drag and in propeller efficiency with altitude).

Figure 4 shows Vx and Vy for the altitudes of sea level, 5,000 feet MSL and 10,000 feet MSL.

At 10,000 feet MSL, Vx is 77 knots (TAS) and Vy is 83 knots (TAS). Here's the most important thing to notice. Vx and Vy (as true airspeeds) both increase with an increase in altitude, but Vx increases FASTER than Vy as shown in Figure 5. Eventually, as altitude continues to increase, Vx will catch up with Vy at the point where the airplane has a "zero" rate of climb. This point is known at the airplane's absolute ceiling.

Why Vy is Said to Decrease With Altitude
So far we've spoken only about true airspeeds. When we talk about indicated airspeed, things change a bit. Here's why.

In order to understand how Vy (as an indicated airspeed) decreases with altitude, we need to convert the true airspeeds from Figure 5 to indicated airspeeds.

As a rule of thumb, we know that our true airspeed increases approximately 2% per thousand feet over our indicated airspeed. In other words, if we're indicating 100 knots at 10,000 feet, then our true airspeed is 20% more than what we indicate, or 120 knots.

But what if I want to know what indicated airspeed is needed to produce a certain true airspeed? You can determine this by using a simple mathematical formula like the one shown in Figure 6. Let's use it to convert the true airspeeds in Figure 5 to indicated airspeeds.

Find the IAS required to produce a specific TAS by using this formula (Figure 6).

Now let's put those values on a chart as shown in Figure 7.

When you convert the true airspeeds to indicated airspeeds for the altitudes mentioned above (S.L., 5,000' & 10,000'), you find that Vy, as an indicated airspeed actually decreases with altitude. Vx, on the other hand, increases with altitude as an indicated airspeed (Figure 8).

Here's another way of looking at these results. Let's assume for a second that Vy, as a true airspeed, stays the same with altitude. If that was so then the indicated airspeed to maintain a single (specific) true airspeed would decrease with altitude. That should make sense since the air becomes less dense, therefore, we move faster through the air but show less of an indicated airspeed reading while doing so. In reality Vy as a true airspeed actually increases with altitude. But the increase in TAS readings is slow enough that each specific IAS required to produce a specific TAS decreases with altitude. You can see this clearly from Figure 9.

Looking at the Vy (bright red) line, notice that both TAS and IAS are assumed to be equal at S.L. At 5,000 feet, it takes an IAS of 73 knots to produce a TAS of 80 knots. At 10,000 feet, it takes an IAS of 69 knots to produce a TAS of 83 knots. Thus, you can see how Vy as an IAS decreases with altitude.

On the other hand, Vx (as a TAS) increases faster than Vy (as a TAS) with altitude. Therefore, the IAS required to produce a specific Vx (as a TAS) will also increase faster that that for Vy. While this is hard to explain verbally, you can see this clearly from Figure 9. The Vx (purple) line shows that both TAS and IAS are assumed equal at S.L. At 5,000 feet, it takes an IAS of 63 knots to produce a TAS of 69 knots. At 10,000 feet, it takes an IAS of 65 knots to produce a TAS of 77 knots. Thus, you can see how Vx as an IAS increases with altitude. Remember, differences between IAS and TAS occur because TAS increases with altitude.

I hope this helps.

Best,
Rod