Flight for range explores the question - how much fuel will the airplane use to get to its destination? Flight for endurance explores the question - how much fuel does the airplane use in a unit of time?
We will define the term Specific Range (SR) to be the ratio of groundspeed/fuel flow. The units of SR are nautical miles per gallon, or nautical miles per pound of fuel. It is better to use units of nautical miles per pound of fuel because the chemical energy of a unit of fuel depends on the mass of the fuel. So, as fuel gets colder and therefore more dense you actually get more energy from a gallon of fuel, but always get the same amount of energy out of a pound of fuel.
We will define the term Specific Endurance (SE) to be the ratio 1/Fuel Flow. The units of SE are hours per gallon, or hours per pound of fuel. The unit hours per pound of fuel is preferred.
The greater the value of SR the further the airplane can fly (for a given amount of fuel onboard.) It is the equivalent of saying that your car gets 40 miles per gallon while someone else's only gets 20 miles per gallon. Obviously the one that gets 40 miles per gallon can go further between fuel stops, or requires less fuel when it does stop to gas up.
The greater the value of SE the longer the airplane can stay in the air. With a car you can always pull over and shut the engine off. In that case fuel flow (FF) becomes zero and SE (1/FF) becomes infinite. In other words your fuel will last forever - because you aren't using any. Unfortunately an airplane must keep moving to stay in the air; therefore you cannot reduce fuel flow to zero. But, if you fly at the lowest possible fuel flow you will have the maximum possible SE and will be able to stay in the air as long as possible.
In small propeller airplanes pilots almost never fly at the speed for maximum range. Pilots tend to fly much faster than ideal, just as we all tend to drive our cars faster than ideal. But, pilots of jet airplanes, especially on long range transoceanic flights, do try to fly at the speed for maximum SR. We will spend considerable time discussing the intricacies of how to do that later.
A pilot would only be concerned about maximizing endurance (SE) if s/he were flying circles. That could happen - for example the runway may be closed for snow removal, or to remove FOD etc. In such a case the pilot might circle overhead trying to consume fuel at the lowest possible rate. In this case we want maximum SE, not SR.
A "true" jet engine takes cold air in through an inlet (left side of diagram to left.) The cold air is compressed as it passes through the compressor section. The compressor consists of a series of rotating blades that work like a propeller, and stator blades that are fixed.
For a subsonic engine the compressor section is shaped as a convergent chamber - i.e. like a venturi that causes the air to accelerate as energy is added to it. Just before the combustion chamber there is an expansion chamber so the air slows down and the pressure rises (as per Bernoulli's equation.) When the air enters the combustion chamber it is compressed and at moderate velocity (well below the speed of sound.)
The air then enters the combustion chamber where fuel is injected and ignited. The air and fuel mixture in the combustion chamber expands and rushes out the back of the combustion chamber at very high speed.
In the turbine section the rapidly moving air flow is used to turn some turbine blades. These work just like a windmill, except the air flowing over them is red hot, so they must be made of tungsten to prevent them from melting.
The turbine blades are connected to a shaft that turns the compressor blades at the front of the engine. Thus the engine is self sustaining. You can see the shaft in the middle of the above diagram.
The turbines remove a small amount of energy from the air, but it is still flowing at high speed when it leaves the outlet at the rear of the engine (right side of above diagram.) Thrust results primarily from the difference in velocity of the air entering the inlet and that exiting the outlet. There is a small amount of additional thrust due to the increase in mass due to fuel being added, but the mass of fuel is only about 4% of the mass of the air worked on by the engine, so most of the thrust results from the air being accelerated.
The above explanation is for a true jet engine, which is an engine in which all the air entering the inlet passes through the combustion section. Modern jet engines are always bypass engines. A bypass engine is shown below. The bypass engine is more efficient because it works on a larger mass of air and therefore doesn't need to change the velocity of the air as much to create thrust.
Thrust Specific Fuel Consumption (TSFC) is defined as the ratio of fuel flow to thrust produced by the engine:
We use the abbreviation Ta to represent "thrust available" the letters FF represent fuel flow, and the units should be pounds of fuel per hour per pound of thrust. (Note: Most small airplane handbooks specify fuel flow in gallons per hour, which can be converted to pounds per hour (approximately) by multiplying by 6.)
For JET engines TSFC is very close to being constant over a wide range of airspeeds. In other words the same amount of fuel flow produces the same amount of thrust at any speed from zero to close to the speed of sound.
We must emphasize the above point: TSFC is very close to being constant for a JET engine. This is a critical point. It means that to determine the fuel flow we simply use the formula: FF = TSFC x Ta. Given that TSFC is a constant this is a very useful and powerful equation. Let's emphasize it again:
FF = TSFC x Ta [for a jet engine - but NOT for a propeller engine.]
Previously we learned that thrust must equal drag in level flight. Therefore we can say that:
FF = TSFC x Drag [for a jet airplane flying level.]
Mass Flow Affects The Value of TSFC (the Bypass Engine)
A great deal of research goes on every year to improve the TSFC of jet engines. Every percentage point improvement in TSFC will save hundreds of thousands of pound of fuel consumed by all the airliners of the world every year. And like they say, a million here and a million there and pretty soon your talking real money.
Even a casual observer will have noticed that jet engines have been getting larger in diameter as the years go by. The main reason is that the larger the diameter of the engine the more fuel efficient it is.
The picture to the left shows a modern Boeing 737-700. Compare the engines on this airplane to the ones on the ProfessionalPilot.ca logo at the top of every page. That airplane is also a Boeing 737, but a much older one; you can see how much smaller the diameters of the engines are.
As already noted, a jet engine produces thrust by accelerating air. The thrust equals the mass of air times the acceleration, this is just Newton's second law:
F=ma - i.e. Thrust (T) = (mass of air) x (acceleration given to the air)
T = m x dV [dV represents change in velocity. Above we saw how a jet engine changes the velocity of the air it works on.]
The velocity of the air entering the inlet is zero (more or less.) The air leaving the outlet is moving much faster. As result the engine leaves the air behind the airplane disturbed (in motion), which represents a loss of efficiency. The more residual motion the air behind the engine has the more energy has been wasted. Therefore, an engine is more efficient if a large mass is given a smaller dV:
The engine to the left is a high bypass jet engine, which means that in addition to the core of the engine, as described above, there is a large "fan" at the front of the engine that accelerates a large mass of air. In the picture the fan is the part surrounded by the brown shroud. The air acted on by the fan bypasses the core of the engine, hence the term bypass engine. This type of engine is also often called a "fan-jet." Fanjets are more efficient than true jets - i.e. they have lower TSFC. All modern jet airplanes are actually fanjets, not true jets.
When a true jet engine accelerates air it does so by injecting fuel into the air and igniting the fuel. The fuel burns which causes the air to accelerate. Two effects are at work; the minor effect is that each hydrocarbon molecule in the fuel brakes down into many individual carbon and hydrogen atoms that combine with oxygen to produce C02 and H2O. The increased number of molecules plus the rise in temperature due to combustion causes a substantial rise in pressure that forces the gas to rush out the back of the engine. Thrust is the reaction force, as per Newton's third law.
The dominant factor in producing thrust is the temperature rise of the air; therefore, the colder the air entering the engine the more efficient the engine will be. In the atmosphere temperature usually decreases with altitude so jet engines are more efficient at higher (colder) altitudes than at lower (warmer) altitudes. Obviously on a cold winter day it can be cold at low altitudes so you should remember that it is temperature not altitude that actually matters. However, see the next point.
As we saw above thrust results from accelerating a mass of air. Given that air density represents the mass per unit volume of air we must expect a jet engine to produce less thrust at high altitudes where the air is less dense. That is definitely the case.
At 22,000 feet the air is about half as dense as at sea level, so a jet engine will produce about half as much thrust there. Jet airplanes must have engines that produce MUCH MUCH more thrust at sea level than will be needed in cruise at high altitude. A side benefit is that all jet airplanes have LOTS and LOTS of thrust available for takeoff and climb. There is however a disadvantage to having such powerful engines; it makes it impossible to cruise efficiently at low altitude. The reason is discussed next.
Jet engines operate efficiently only when operated at close to maximum rpm. If you throttle a jet engine back below a critical rpm it becomes very inefficient. It is important for pilots to know that.
The tachometer on a jet engine is usually calibrated in percent (%.) The pilot needs to keep the engine at a high rpm, more than 80% at least, and closer to 100% is better, for it to operate efficiently. If s/he throttles back the TSFC will start to rise.
In the diagram to the left a business jet airplane (W = 50,000 lb) has just taken off at sea level and is climbing at 350 knots. The engine is operating at 100%. The horizontal red line represents the thrust the engines are producing at 100%. The engines are producing 15,000 pounds of thrust. We can see that there is much more than enough thrust for level flight (drag is only 5495 lb.) If the pilot leveled off now s/he would have to throttle back to less than 50% rpm ( and that's a bad thing.)
In the diagram to the left the same airplane has reached 36,000 feet and the pilot is about to pitch the nose over and accelerate to cruise speed of 500 KTAS.
The engine is still operating at 100%, but in the much less dense air at 36,000 feet the engine is now producing only 4470 pounds of thrust (see above for reason why.)
Drag is 3670 pounds, but will increase somewhat when the airplane accelerates to 500 knots (what will it be?)
In the left margin of the diagram the drag in cruise at 500 knots has been marked. You can see that the pilot will not have to pull the throttle back very much. So, the engine will continue to operate just below 100% and as such will be very efficient. I.E. TSFC will be "good."
Imagine what would have happened if the pilot had chosen to level off at a lower altitude. S/he would have been forced to reduce the throttle substantially to match thrust to drag. The result would have been a power setting much less than 100% and the TSFC would have been "poor." It should be clear that the designer of the airframe and the engine must work together so that both will work optimally together at cruising altitude.
In summary, the amount of thrust produced depends on the density of the air. In above example 100% rpm produces 4470 lb of thrust at 36,000 feet. At sea level the pilot would have to throttle back to less than 50% to produce the same 4470 lb of thrust - and the engine would not be efficient.
NOTE: you can recreate the situation shown above, using the climb performance simulation, and explore it more completely. We will get to that simulation later in this section.
We are now almost ready to analyze range and endurance performance of a jet airplane. What we need is a graph relating fuel flow to velocity. Since we know that fuel flow is directly related to thrust and we also know that thrust equals drag the process is quite simple. We start with a drag vs. velocity graph and convert it into a fuel flow vs. velocity graph.
In the early days of jet engines the value of TSFC was pretty close to 1.0. In other words 1000 pounds of thrust required 1000 lb/hour of fuel flow, or 20,000 pounds of thrust required 20,000 lb/hour of fuel flow, etc. In such a case you can just take a Drag vs. Velocity graph and re-label it as Fuel Flow vs. Velocity with no further changes.
Thankfully modern bypass engines are more efficient, but still TSFC is usually about 0.8. In other words 1000 pounds of thrust requires a fuel flow of 800 lb/hour of fuel, or 20,000 pounds of thrust requires a fuel flow of 16,000 lb/hour of fuel.
In the simulation below you can set the value of TSFC and experiment to see how it affects range and endurance.
The process of creating a Fuel Flow vs. Velocity graph is really very simple; you simply take a Drag vs. Velocity graph and adjust the y-axis to represent Fuel Flow by using the definition of TSFC - i.e. FF - Drag x TSFC.
The simulation creates a Fuel Flow vs. Velocity graph for both a jet and a propeller airplane. We discuss propeller airplanes later in the course, so you can ignore it for now. To hide the propeller airplane curve click the "Hide Curve" button in the propeller airplane information box.
The simulation works by first calculating the drag. You can display the parasite drag and induced drag curves if you like, although they are hidden by default. The simulation is for level flight only, and we know that thrust equals drag in level flight. So, the simulation simply takes all the drag values and multiplies them by TSFC. The result is fuel flow, and this is presented on the y-axis.
NOTE: a different process is required for the propeller airplane and we will cover that later. For now ignore the propeller airplane except for comparative interest.
You can set TSFC. (You can also set SFC, which is for the propeller airplane. We discuss SFC later.)
To operate the simulation use the up and down arrow keys to change AOA. As you change AOA the airplane speeds up and slows down, but it always remains in level flight.
At the lower left corner you can modify the design of the airplane. To change the aspect ratio drag the green navlight in or out. To change wing area drag the "S" symbol up or down. Similarly change weight by dragging the "W" symbol up or down.
Use the E and C key to change density altitude. If you use the shift key you can change altitude in 10,000 foot increments. Without the shift key altitude changes in 1,000 foot increments. You can go all the way up to 70,000 feet, but remember that the engine loses thrust in proportion to air density.
Drag the windsock to change the wind. The simulation properly calculates the best range speed taking the wind into account. Our next task is to analyze what speed to fly for maximum range or endurance. That discussion is below the simulation.
Please NOTE that the simulation does NOT take supersonic effects into account. Every airplane has some critical speed above which drag rises due to shockwave formation. We discuss that later. In this simulation there is no allowance for shockwave drag so if you fly at speeds more than about M=0.85 you are probably fooling yourself about the efficiency of the airplane.
In the simulation the y-axis represents FF. Clearly maximum SE occurs at the lowest possible FF - recall that SE = 1/FF. The lowest possible fuel flow occurs at the bottom of the Fuel flow vs. Velocity graph. Since fuel flow is proportional to drag it is also true to say that minimum fuel flow occurs when the airplane flies at L/Dmax. We say that maximum SE occurs at L/Dmax.
In the drag chapter we learned that L/Dmax occurs at a particular angle of attack. Therefore, the actual speed the airplane must fly at for maximum SE decreases as fuel is burned and the airplane gets lighter.
In the drag chapter we also leaned that the total drag curve moves to the right as altitude increases but L/Dmax does not change with altitude. This leads to a preliminary conclusion that altitude will not affect endurance. We will explore this conclusion more in a moment.
You should note that maximum endurance speed for a jet airplane is the same as maximum glide speed. Both occur at the angle of attack for L/Dmax.
How does altitude affect endurance of a jet airplane?
Using the simulation you can see than minimum drag is NOT affected by altitude. Therefore, aerodynamically altitude has no affect on endurance. However we discussed earlier that the jet engine must be operated at high rpm to be efficient. And we also noted that colder air temperature improves TSFC. Therefore, if you have a choice you would fly at altitudes in the stratosphere where the air is coldest and you can keep the engine rpm high to achieve maximum SE.
How does wind affect endurance of a jet airplane?
Wind affects groundspeed. But when we fly for maximum endurance we are just flying circles so groundspeed does not matter.
Interesting Point: If you do get caught at low altitude and need to maximize SE (minimize fuel flow) in an emergency there is a trick you can use if your airplane has four engines rather than the usual two. Simply shut down two of the engines. Once you do that you can operate the remaining two at high rpm while flying at L/Dmax angle of attack. This may strike you as a crazy idea - and I don't seriously recommend it to B747 pilots. But it is actually done in some military operations.
Imagine that you are flying at L/Dmax, then start to reduce AOA below the L/Dmax AOA; the fuel flow increases, but so does airspeed. Since SR = TAS/FF range actually improves initially. Eventually fuel flow starts to rise so rapidly that SR decreases again.
Remember that FF = T x TSFC.
Remember that SR = Speed/FF. For now let use use TAS as the speed value, but it really should be groundspeed that we use. We will deal with that later.
Therefore, in zero wind SR = TAS/(T x TSFC).
In level flight TAS is tied to the CL1/2 as we saw in the Lift chapter, and Thrust is proportional to CD. Therefore we conclude that SR will be proportional to the square root of CL divided by CD or in other words:
The maximum SR will occur when is at its maximum value.
On the Fuel Flow vs. Velocity graph the point for maximum SR can easily be found graphically. Consider the diagram to the left.
Chose any point on the graph then draw a line from the origin to that point, as shown in the diagram.
Now consider the angle "R" between the line just drawn and the X-axis. What is the value of Tangent(R)?
tan(R) = FF/TAS [examine the diagram to convince yourself of this]
Recall that the definition of SR is SR = TAS/FF. In other words:
SR = 1/tan(R)
Summary. The tangent of any point on the FF vs Velocity graph equals the SR for that point. Now we ask the question - where is SR maximum?
SR must be a maximum where tan(R) is a minimum.
The point for maximum range is easily found by inspection. The point with the minimum tangent of R is shown in the diagram to the left.
If you got lost in the mathematical development above simply realize that this point is the one for which the ratio of speed to fuel flow is maximized.
We already know that flying at the point shown to the left corresponds to flying at the angle of attack for .
In the simulation the computer calculates this angle of attack for you. Notice that if you fly at this angle of attack you always get maximum range.
Next we will explore how weight and altitude affect SR.
In the simulation the default weight for the airplane is 2400 lb, which is very unrealistically low for a jet. We will try increasing the weight and see what changes. Before we do we should record the starting figures. occurs at AOA = 2.1 degrees (if you changed the default AR change it back to 7.3 and set CDp = 0.024) Change AOA as needed, using the up and down arrow keys, to 2.1°.
The "X" that represents your current speed and fuel flow should now be at the point where the tangent line intersects the curve. Make sure the tangent line is visible, but hide the propeller curve to avoid clutter. Your graph should look like the one to the left.
Record the data in the jet data box:
Now we will increase the weight to that of a small personal jet with a wing loading of 60 /b/ft2. To do that increase W to 10,445 lb.
The resulting graph is shown to the left. The CRUCIAL first thing to notice is that maximum range still occurs at AOA = 2.1°. You may be surprised, but you shouldn't if you remember our previous analysis that showed that:
We previously said that L/D ratio depends on only three factors, which are expressed in the above equation:
Because of the way the simulation adjusts the y-axis to keep the graph optimally viewable it may be difficult for you to fully visualize how and why the fuel flow curve changes with weight. The graph to the left should help.
In this graph the y-axis has been kept constant as the weight changes from 2400 lb to 10,450. Check the values against the graphs above to convince yourself that they are the same values.
The parasite drag and induced drag curves are visible in this graphic.
As we expect parasite drag does NOT change when weight changes.
As we also expect, induced drag increases when weight increases. The graph shows induced drag at 2400 lb, 6200 lb and 10,450 lb.
The drag curve, and therefore the FF vs Velocity curve must always shift up and to the right as weight increases. It is actually better to think of it as rolling up from the lower left as induced drag increases but parasite drag does not change.
We can easily see that the L/Dmax point is higher up at higher weight. So just as common sense would predict the heavier airplane will consume more fuel and have a lower SE.
The tangent lines drawn from the origin to the point ("X" point) have been left out to avoid clutter, but we can easily see that if we drew them in SR would get worse as W increases.
In summary, we know that an airplane always achieves maximum endurance at the L/Dmax AOA, which does NOT change with weight. But the actual value of SE does decrease at higher weight. We also know that maximum range always occurs at the AOA, which does NOT change with weight. But the actual value of SR decreases as weight increases. Both of these effects are pretty much what common sense would have predicted.
To understand how altitude affects range and endurance of a jet airplane you must recall what we previously learned about how drag changes with altitude. We learned that parasite drag decreases with altitude. This was fairly intuitive since we expect the less dense air to cause less drag. We also learned that induced drag INCREASES with altitude. That is because the airplane must fly at a higher angle of attack for a given TAS at the higher altitude.
It is often easier to simply remember that both induced and parasite drag are always the same for a given EAS. So, at higher altitude where TAS > EAS the drag curve shifts to the right. If we ignore the effect of temperature on TSFC for a moment then the FF vs Velocity curve will mirror the changes in the drag curve. The graph below shows FF vs Velocity at sea level and at 45,000 feet for the personal jet (W=10.450) mentioned above.
The graph shows the induced and parasite drags at both sea level and 45,000'.
The parasite drag decreases and the induced drag increases with altitude. The result is that the drag curve moves to the right.
Since FF = TSFC x Drag the FF curve also moves to the right.
The horizontal green line shows that minimum fuel flow does not change with altitude - IF TSFC does not change. Remember that we said earlier that TSFC will decrease when temperature decreases. To simulate that when you use the simulation above you should manually reduce TSFC as altitude increases. Remember that temperature only decreases up to the tropopause (36,000') then becomes constant.
Based on the above we would expect a jet to achieve maximum endurance at the tropopause or above (i.e. in the stratosphere.)
The above graph makes it pretty clear that altitude is NOT much of a factor in maximum endurance for a jet. However, it is equally clear that altitude is a MAJOR FACTOR in range.
As we learned previously maximum range always occurs at the AOA for . The corresponding points are marked with "X" on the graph.
We also learned that SR = 1/tan(R) where R is the angle as labeled in the graph above. It is quite clear that the angle R gets less and less as altitude increases. In other words SR becomes better and better as altitude increases. NOTE that this improvement is in addition to any improvement due to temperature drop. Therefore, SR continues to increase as the jet climbs into the stratosphere (unlike SE which becomes constant.)
Please NOTE that while the above analysis indicates that SR will become infinite if you climb to infinity there is a practical limit - actually two practical limits. The first limit is engine thrust. We already learned that jet engines produce less thrust at altitude. So, the engine will run out of thrust at some altitude and the pilot won't be able to climb any higher. The second practical limit is the speed of sound. The graph shows that the true airspeed the airplane cruises at gets greater and greater at higher and higher altitudes. At some altitude the airplane will be going so fast that shockwaves will begin to form. This will increase parasite drag and invalidate the assumptions that underlay our analysis. We now turn to a more detailed analysis of this point.
On long range flights a jet pilot practices a technique known as cruise control. Its a bit more complicated than using a cruise control button on a modern car - but not much.
To understand cruise control you only need to know two things. One we have reviewed over and over, the other is a point we have only implied:
The objective of cruise control is to fly BOTH at and maximum Mach number.
To understand cruise control follow through the following example using the simulation. We will assume we jet transport pilots - as such setup the simulation with the following data:
Once you have set the above values you will notice that you need to fly at AOA = 2.9° to achieve maximum SR - use the up and down arrow keys to go to that AOA.
Notice that you are only flying at TAS = 292 and you Mach number is only 0.44 (values are shown on the airspeed indicator.)
Now let us assume that this airplane has a maximum Mach number of 0.85. What can we do to get the TAS up to Mach = 0.85 while remaining at our present angle of attack (2.9°)?
The only thing we can do is climb. Start a climb - you will see that your EAS remains at 293, but your TAS and mach number increase with altitude (Mach number = TAS / speed of sound) When you reach 32,000 feet you will be cruising at Mach = 0.85. STOP your climb here. If you go higher you will have to increase AOA to stay below the Mach limit, but if you had leveled off lower you would either have been flying slower than Mach 0.85, or would have used an AOA less than .
You should be able to see that there is just ONE ALTITUDE at which we achieve our objective of flying at both and maximum Mach number.
What happens as the flight progresses and we burn fuel?
To answer this question reduce W to 700,000 - but don't change any other airplane parameter.
The reduction in weight reduced the wing loading from 150 lb/ft2 to 131 lb/ft2. The simulation is programmed to stay at the same AOA, so AOA is still 2.9° as desired. But, the EAS has dropped to 273 KEAS and Mach number is down to 0.79.
We have just learned that as fuel is burned the pilot MUST slow down to stay at the optimum angle of attack. But, is there any way we can keep our Mach number up to the desired Mach 0.85?
Yes - try climbing. You will find that if you climb to 35,000 feet you can regain Mach 0.85. You have just made your first STEP CLIMB - welcome to the world of jet pilot aviation.
In cruise control the pilot makes a series of step climbs as fuel is burned. That way s/he can keep the airplane at and maximum Mach number. Apparently sometimes you can have your cake and eat it too :)
The above system is called cruise control.
Previously I mentioned that when we calculate SR we should use:
SR = groundspeed / FF.
Groundspeed is just TAS +/- headwind or tailwind. It should be obvious that SR is better with a tailwind and poorer with a headwind. But should we still fly at ? The answer is NO!
In the simulation you can change the headwind/tailwind by dragging the windsock icon at the lower right corner of the graph.
To graphically allow for the headwind simply realize that we need to subtract the headwind from the TAS to get groundspeed. On the graph to the left you can see how that is done.
In this example there is a 100 knot headwind. To plot this graphically we start the tangent line at 100 knots that way the speed value is 100 knots less than if we started from the origin.
The tangent line touches the FF vs. Velocity curve at the revised speed for best range.
You can see that the angle "R" is greater than it would be if the line was drawn from the origin. Thus SR is worse with a headwind, which is what common sense should have told us.
The computer calculates the best angle of attack for SR with the wind and without. This information is displayed in the Jet A/C data box, as shown to the left.
The airplane in this example is the same jet used in the cruise control example above. We saw previously that it must fly at 2.9° AOA for maximum SR in zero wind. But with the 100 knot headwind it must be flown at 2.1° AOA. This corresponds to changing airspeed from 273 KEAS to 300 KEAS. (Note that using the same cruise control concept discussed above the pilot must descend to 31,000 feet, due to the higher required EAS.)
It is important to realize that we must always fly faster when flying into a headwind. Ideally we should slow down slightly with a tailwind. The same graphical method can be used to find the optimum speed with a tailwind. Try it in the simulation above to see.