June 2005 Article of the Month
Why Vx and Vy Change With Altitude
Dear Mr. Machado
I have been flying for a while but have started to read (and enjoy a great deal) your work only relatively recently. I was particularly impressed by the clarity of your recent explanation of maneuvering speed Va, which has prompted me to suggest another subject: During flight training we have all been taught that Vx increases and Vy decreases with altitude, converging at either the "service ceiling" or "absolute ceiling" of the aircraft--I forget which. 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.
Thanks,
K.M.
Greetings Mr. K:
Thanks for those wonderful comments. Your question about Vx and Vy is a good one. In fact, it's one of the more difficult things to explain. Any explanation requires the association of several ideas to adequately answer the question. Here's my attempt.
First, we'll plot everything on a graph using true airspeed since it's true airspeed that changes with altitude, not indicated airspeed.
So let's first start with why Vx increases with altitude.
As you know, Vx is the best angle of climb speed. At this speed the airplane acquires the most altitude (vertical movement) for a given distance (horizontal movement) over the ground. Take a look at Figure 1 below.
The red 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 top of the curve, you arm would point in the same direction as the tangent line shown in the figure. The angle your arm makes with the horizontal represents the maximum angle you can point upward and still touch the rate of climb curve (ROC) at point A. Thus, mathematically speaking, the maximum upward angle relative to the horizontal, represents the maximum "angle" of climb for your airplane. If you drop straight down from point A, you'll find point B, which represents an airspeed of 69 knots. Therefore, our best angle of climb speed at sea level is 69 knots TAS (which is also 69 knots TAS since we're at sea level and there is no practical difference here between TAS and IAS)..
Now, ask yourself what happens to the ROC curve when altitude increases. Since power decreases with altitude, the ROC curve will change shape, grow shorter and move to the right as shown in Figure 2. Naturally, this is the result of the airplane's ROC decreasing at higher higher altitudes.
If you apply the same logic, it stands to reason that a shorter ROC curve causes the tangent line to make less of an angle with the horizontal. You can think of this as an airplane climbing at a geometrically smaller climb angle relative to the horizontal as altitude increases. Therefore, dropping straight down from the tangent point (point A) to point B, we find that the best angle of climb speed has increased to 90 knots TAS (which, is actually 75 knots IAS). Remember that your true airspeed (TAS) increases 2% per 1,000 feet over the value of indicated airspeed (IAS). That means that a TAS of 90 knots at 10,000 feet is actually an IAS of 75 knots. (Here's the math just in case this isn't clear to you: At 10,000 feet, our IAS is 75 knots knots, which means our TAS is 20% greater. Take 20% of 75, which is 15. Add this to 75. This equals 90.)
It's should now be clear that Vx increases from an IAS of 69 knots to an IAS of 75 knots at 10,000 feet. Describing it this way sure beats trying to explain the same thing using differential calculus.
Let's move on to why Vy changes with altitude.
Vy is the best rate of climb speed. It's the speed that results in the greatest upward deflection on the VSI. Since an airplane's rate of climb is a function of excess power, it stands to reason that the more excess power you have the higher the rate of climb will be. Take a look at Figure 3 below.
The power required curve (PR) shows the power required to maintain level flight. It's curved because of the effects drag. Remember, induced drag increases as the airplane slows and decreases as it speeds up. Parasite drag, on the other hand, decreases as the airplane slows and increases as it speeds up. The power available (PA) curve is also curved because propeller efficiency changes at different airspeeds. After all, the propeller is nothing more than a moving wing isn't it? Therefore, it experiences its own drag effects which are similar to the drag experience by the airplane's wings.
There is a point between the PA and PR curves where maximum separation exists (the vertical red line). This is the point where you have the minimum power required for level flight and the maximum amount of excess horsepower capable of being produced by the engine. Following this point down to the bottom of the graph tells you the airplane's best rate of climb speed. In this example, Vy is approximately 82 knots true airspeed (TAS), which is also 82 knots IAS since this airplane is at sea level.
Now ask yourself what happens to the PA and PR curves as altitude increases. Yes, they also change shape and position on the graph as shown in Figure 4.
Notice how the curves came closer together and moved slightly to the right. This should make sense since the power available (PA) curve must move closer to the bottom of the graph with an increase in altitude (we're assuming the airplane isn't turbocharged). The power required (PR) curve moves up and shifts to the right slightly. Find the point of maximum separation between the two newly shifted curves (the vertical red line). Move straight down to the airspeed line and you'll see that Vy has actually INCREASED with an increases in altitude (it's now 90 knots, TAS). In fact, Vy increases approximately 1% in value per 1,000 feet altitude increase. Now hold on for a second because I realize that this appears contradictory to what you've heard about Vy. In reality, it isn't contradictory. Let me explain why with two basic observations.
OBSERVATION # 1.
If Vy is measured as a true airspeed, it does indeed increases approximately 1% for every thousand feet increase in altitude. Therefore, if Vy is 82 knots (TAS) at sea level, it must be approximately 90 knots (TAS) at 10,000 feet. (Here's the math: 10% of 82 is 8.2, so add 8.2 to 82 and you get an approximate value of 90 knots.)
Now ask yourself what IAS is necessary to give you a TAS of 90 knots at 10,000 feet? How do you figure this out? You work backwards from the old rule of thumb you were taught as follows in observation #2.
OBSERVATION #2
As mentioned above, TAS increases 2% per thousand feet for any given IAS. At 10,000 feet, our Vy is 90 knots, which is equal to 75 knots IAS. Therefore, as an indicated airspeed (IAS) our Vy has decreased from a sea level value of 82 knots to a value of 75 knots at 10,000 feet MSL.
That's it. You're finished with the problem. Pretty easy right?
Even though these Figures don't represent a specific airplane you can see that at 10,000 feet Vx equals Vy (both are 75 knots IAS). Therefore, we can say that, for this fictitious airplane, 10,000 feet is the airplane's absolute ceiling.
Whew! That's a lot. If you stick with it you'll eventually understand the explanation. It's as simple as I can make it without having to disregard the science behind the effect. I hope this helps.
Best,
Rod