I will admit that I am a recovering engineer turned pilot who still enjoys a good differential equation or two. But I am recovered enough to acknowledge you don't need anything more than basic math to fly an airplane. But if you want to really understand your craft, you need to know something about aerodynamics. And all that begins with the purest form of physics: the field of mechanics.
Sir Isaac Newton — The Patron Saint of any engineer who hasn't given in to the "Relativity Fad"
Eddie contemplates motion on two wheels
First things first: we need to discuss basic position in space. Starting with two dimensions, we can measure our position forward with positive numbers, backwards with negative numbers. In the drawing, the bike is at 0.
If after 1 minute the bike has traveled 1 mile, we can compute the speed thusly:
Or, preferring a more standard measure:
Of course the speed you see on the speedometer isn't necessarily the speed you will accomplish over the duration of the trip. To figure average speed, you need a reference point at the start and at the end of the time in question. An odometer makes as good a reference as any, say you start the trip with the odometer reading 001000 miles.
If after an hour the odometer reads 001060, your average speed has been:
But let's say you run into traffic and after two hours the odometer reads 001090, your average speed has fallen to:
So far we've discussed speed, which is a scalar quantity. That is, it has a magnitude but no direction. As pilots, we are interested in velocity, which is a vector quantity. It has magnitude and direction.
You can think of direction as angles on a grid, or more conveniently, angles on a compass. We can say in our drawing the motorcyclist is heading east-north-east, 30° north of east, or best of all, heading 060.
Where speed is simply a number, to the physicist velocity is an arrow. The length of the arrow shows magnitude, the direction shows, well, direction. Sure, to a pilot velocity is a number. But always keep in mind there is a direction associated with that number.
From all this we can derive a basic equation for velocity:
Through the magic of algebra:
Before we go any further, let's talk about units. Miles versus feet, hours versus seconds, it can be a real mess. It doesn't matter what you use, but when the units you choose must be consistent in every equation. So for the sake of example let's say you are going 50 mph, miles per hour. To convert that into feet per second you need only multiply it by 1 set to units of your choosing. 1 hour divided by 60 minutes, for example, is equal to one. You can multiply your starting number with these conversion factors, just so long as all the units are multiplied through as well. In the example, the hours, minutes, and miles all "cancel" out to reveal the answer: 84 feet per second.
Velocity on a Graph
You can also plot distance versus time to come up with a graphical representation of velocity. The slope of the distance change along any point of the line will show the velocity over that time span. Since our velocity is constant — neither accelerating or decelerating — any two points will yield a velocity of 84 fps.
We define acceleration as the change in velocity per unit of time. If, for example, we go from 0 mph to 50 mph in 10 seconds, we have a mathematical mess on our hands. Converting the velocity into feet per second, as we saw above, makes things more sensible.
The "feet per second square" is a standard unit of acceleration
If we establish the symbol Δ, delta, as a change in values, Δv denotes the change in velocity. Ergo:
Acceleration on a Graph
Just as you can also plot distance versus time to come up with a graphical representation of velocity, so to can you plot velocity versus time to come up with a graphical representation of acceleration. The slope of the velocity change along any point of the line will show the acceleration over that time span. Since our acceleration is constant any two points will yield an acceleration of 8.5 fps2.
If a = v / t, and we know that it is, we can do some algebra and find that v = at. Using that, we find that the average velocity of something accelerating at a constant rate a is (0 + at) / 2. Well, zero is an easily eliminated term isn't it?
But we already know d = vt, so that means d= (½at) t, so:
Jerk is a subject for sophomore physics to provoke sophomoric laughter from the back of the class, but it is a legitimate measure of mechanics that is especially pertinent to pilots, be they jerks or not. How does the motorcycle rider experience jerk? Perhaps we should go back to the beginning.
Sitting on the bike with the motor off and two feet planted on the ground, the rider has a velocity of zero. Because there is no change in velocity, acceleration is also zero.
Cruising along at a steady 50 mph, the velocity can be said to be 84 fps. Because the speed is constant, the acceleration is again zero.
If the rider spots a squirrel on the road and decides to brake, the velocity drops off. If the rider is smooth on the brakes, the acceleration might be constant. Let's say the rider goes from 84 fps to a full stop in 30 seconds. The acceleration is:
Note that we say the acceleration is negative in this instance, it is a deceleration.
If the acceleration is constant, then jerk = 0. Let's say the rider goes from moderate braking, around -2 fps2, to harder braking, around 5 fps2, in the last 5 seconds:
You won't find a jerk meter on the bike (or in your cockpit), but you will know it when you feel it. (Jerk is usually accompanied by irregular head bobbing.)
Jerk on a Graph
Now let's say our intrepid rider underestimates the amount of braking required to avoid the rodent. If he or she then decides to really clamp down on the brakes, then backs off, and then on again, the stop will be said to be "jerky." Or in my case, when I do this with the lovely Mrs. Haskel on back, I will have been said to be a jerk.
Why is this especially pertinent to a pilot? There are lots of reasons covered in the story Smoothness, but here's a sample. You can stop the airplane on a short runway using aggressive braking by applying your best guess of brake pedal and then holding the brakes steady. As the brakes heat up they become more effective and the deceleration will increase. But it will be done smoothly. Or you can apply some brake, a little less, a little more. If Mrs. Haskel is on board, you will be the jerk pilot up front.
Cartoon: Galileo, from You Want Proof?.
Though it isn't something we pilots like to consider, the field of mechanics is concerned with the act of falling. Galilei Galileo, 1564 - 1642, was able to hypothesize that absent air resistance, all objects fall with the same acceleration regardless of mass. He did this with the clever use of ramps, but tests in vacuum chambers confirm that he was right.
Over the years we've come to realize that objects on earth fall at 32 ft/sec2 which is 9.8 m/sec2 for most of the world. When speaking of acceleration to earth — falling —, we substitute the term g for a so that:
Since Galileo went through all that trouble to invent gravity for us, we should apply this knowledge to our discussion of mass . . .
[Dole, pg. 2] Mass is a measure of the amount of material in a body. Weight, on the other hand, is a force caused by the gravitational attraction of the earth, moon, sun, or other heavenly bodies. Weight will vary, depending upon where the body is located in space. Mass will not vary with position.
This mass unit is called the slug.
[Dole, pg. 5] Moments are measured by multiplying the amount of the applied force F by the moment arm l.
The moment arm is the perpendicular distance from the line of action of the applied force to the center of rotation. Moments are measured as foot pounds (ft-lbs) or as inch pounds (in.-lb). If a mechanic uses a 10 in. long wrench and applied 25 lb of force, the torque on the nut is 250 in.-lb.
The aircraft moments that are of particular interest to pilots include: pitching moments, yawing moments, and rolling moments. Some of these moments may oppose each others in direction. It is therefore necessary to classify each moment, not only by its magnitude, but also by its direction of rotation. In the case of pitching moments, a nose-up or nose-down classification seems appropriate. Mathematically, it is desirable that moments be classified as positive (+) or negative (-). For example, if a clockwise moment is considered to be a + moment, then a counterclockwise moment is considered to be a - moment. By definition, aircraft nose-up pitching moments are considered to be + moments.
[Dole, pg. 6] There are two requirements for a body to meet to be in a state of equilibrium:
- There must be no unbalanced forces acting on the body. This is written as the mathematical formula Σ F = 0. Σ (cap sigma) is the Greek symbol for "sum of."
- There must be no unbalanced moments acting on the body. Mathematically, Σ M = 0.
Moments at the fulcrum in [the figure] are 50 ft-lb clockwise and 50 ft-lb counterclockwise. So Σ M = 0. To satisfy the first condition of equilibrium, the fulcrum must press against the seesaw with a force of 15 lb. So, Σ F = 0.
Newton's Laws of Motion
Newton's First Law
"A body at rest will remain at rest and a body in motion will remain in motion, unless acted upon by an unbalanced force."
The first law implies that bodies have a property called inertia. Inertia may be defined as the property of a body that results in its maintaining its velocity unchanged unless it interacts with an unbalanced force. The measure of inertia is what is technically known as mass.
There are several points to be made about this first law that will make the laws that follow easier to, well, follow:
- When there are no external forces, objects move with constant velocity.
- Inertia is resistance to changes in motion.
- The amount of inertia an object has is measured by its mass; more mass means more inertia.
Newton's Second Law
"If a body is acted upon by an unbalanced force, the body will accelerate in the direction of the force and the acceleration will be directly proportional to the force and inversely proportional to the mass of the body."
Acceleration is the change in motion (speed) of a body in a unit of time.
The amount of acceleration a is directly proportional to the unbalanced force F and is inversely proportional to the mass m of the body. These two effects can be expressed by the simple equation:
or more commonly:
Don't forget that g is a for objects on earth falling to earth. That is weight, isn't it? Yes it is:
Newton's Third Law
"For every action force there is an equal and opposite reaction force."
A force is a push or a pull tending to change the state of a motion of a body.
The forces on a G-450 in unaccelerated flight are shown. (Of course this is a simplified concept that ignores the concept of aerodyamic force. For the real story, see: Lift.)
[Dole, pg. 8] In physics, work has a meaning different from the popular definition. You can push against a solid wall until you are exhausted but, unless the wall moves, you are not doing any work. Work requires that a force must move an object in the direction of the force. Another way of saying this is: "Only the component of the force in the direction of movement does any work."
Consider for a moment a metal ball traveling at a constant speed on a flat surface approaching a dip, as shown in the drawing. As we did in freshman engineering class, we say the ball and the surface are both frictionless and there is no air resistance for the sake of this academic exercise.
We know that as the ball goes down the dip in the surface, its speed will increase until it gets to the bottom where it will trade that extra speed to climb the ramp. When it ends up back at the original level, since we are discounting the effect of friction, it will again be at its original speed.
Before it hits the slope, the energy of the ball can be described as the Kinetic Energy (KE) that comes from its motion and the Potential Energy (PE) due to its higher elevation.
[Dole, pg. 8] Energy is the ability to do work. There are many kinds of energy: solar, chemical, heat, nuclear, and others. The type of energy that is of interest to us in aviation is mechanical energy. There are two kinds of mechanical energy. The first is called potential energy of position, or more simply, potential energy (PE). No movement is involved in calculating PE. A good example of this kind of energy is water stored behind a dam. If released, the water would be able to do work, such as running a generator. PE equals the weight W of an object multiplied by the height h of the object above some base plane:
[Dole, pg. 8] The second kind of mechanical energy is called kinetic energy (KE). As the name implies, kinetic energy requires movement of an object. It is a function of the mass m of the object and its velocity V:
[Dole, pg. 8] The total mechanical energy TE of an object is the sum of its PE and KE:
[Dole, pg. 8] The law of conservation of energy states that the total energy remains constant. Both potential and kinetic energy can change in value, but the total energy must remain the same; "Energy cannot be created or destroyed, but can change in form."
[Dole, pg. 9] Power is defined as "the rate of doing work":
But distance / time = velocity, so:
James Watt defined the term horsepower as 550 ft-lb/sec, therefore:
If the velocity is measured in knots rather than feet per second, the formula becomes:
There are two basic types of friction: static and kinetic. If you try to push an object across a table, at first it resists with great force until it breaks free, and then it takes less force to continue pushing it. Static friction exists when the objects are stationary and varies up until the point it gives in. Kinetic friction occurs when the objects are moving and is less than static friction.
[Dole, pg. 10] If two forces are in contact with each other, then a force develops between them when an attempt is made to move them relative to each other. This force is called friction. Several factors are involved in determining friction effects on aircraft during takeoff and landing operations. Among these are runway surface material, condition of the runway, tire material and tread, the amount of brake slippage, and others. All of these variables determine a coefficient of friction μ (mu). The actual braking force Fb is the product of this coefficient μ and the normal (squeezing) force between the tires and the runway:
where Fb = braking force
μ = coefficient of friction
N = normal force on wheels.
Portions of this page can be found in the book Flight Lessons 1: Basic Flight, Chapters 1, 2, and 8.
14 CFR 25, Title 14: Aeronautics and Space, Federal Aviation Administration, Department of Transportation
Air Training Command Manual 51-3, Aerodynamics for Pilots, 15 November 1963
Coram, Robert, Boyd: The Fighter Pilot Who Changed the Art of War, Back Bay Books, New York, NY, 2002
Connolly, Thomas F., Dommasch, Daniel 0., and Sheryby, Sydney S., Airplane Aerodynamics, Pitman Publishing Corporation, New York, NY, 1951.
Davies, D. P., Handling the Big Jets, Civil Aviation Authority, Kingsway, London, 1985.
Dole, Charles E., Flight Theory and Aerodynamics, 1981, John Wiley & Sons, Inc, New York, NY, 1981.
FAA-H-8083-15, Instrument Flying Handbook, U.S. Department of Transportation, Flight Standards Service, 2001.
Gulfstream G450 Airplane Flight Manual, Revision 35, April 18, 2013.
Gulfstream G450 Aircraft Operating Manual, Revision 35, April 30, 2013.
Hage, Robert E. and Perkins, Courtland D., Airplane Peformance Stability and Control, John Wiley & Sons, Inc., 1949.
Hurt, H. H., Jr., Aerodynamics for Naval Aviators, Skyhorse Publishing, Inc., New York NY, 2012.
Technical Order 1T-38A-1, T-38A/B Flight Manual, USAF Series, 1 July 1978.
"You Want Proof? I'll Give You Proof!" More Cartoons from Sydney Harris, Sydney Harris, 1990