Do pilots use the quadratic formula?

Quadratic formulaWe received this question from a high school student in Florida:

…Recently, my Algebra 2 teacher assigned a project to see if men and women in important professions use algebra in real life applications. In our math book it claims that, “Firefighting pilots can use the Quadratic Formula to estimate when to release water on a fire.” My question to you is do you really use the Quadratic Formula to put out a forest fire? If not, how do you do it? Thank you very much for any help you are able to provide and have a great day!

Let’s hear from some pilots.

  1. When you’re dropping water or retardant do you silently solve the quadratic equation in your head? If not, how do you determine when to release the load?
  2. Are there any occasions in your day to day work activities when you use algebra?

In case you want to brush up on the quadratic formula: Wikipedia.

14 thoughts on “Do pilots use the quadratic formula?”

  1. Pilots use very little math, despite flying being a technical job. When flying fires, we do calculate the weight of the aircraft and we need to know our takeoff performance; this is mostly simple addition, multiplication and division, and the use of pre-printed charts. When performing drops, the drop height and release point is a function of feel and experience, and must be adjusted for wind, terrain and surface fuels, etc. In a nut shell, when I approach the drop, it’s all based on what looks right, and on how the previous drop worked out, if I’m following other aircraft. After each drop, we get an evaluation as to drift, drop height, effectiveness, timing, etc, and we use that information to adjust our next drop. If I know my drop started early because of a headwind, I’ll wait just a little longer before releasing the load on the next drop, etc. Other than that, the most complicated we get in math is time, distance, and speed when calculating getting to and from the fire, and fuel burn and time, when calculating how much endurance we have in our fuel tanks.

  2. the most complicated we get in math is time, distance, and speed when calculating getting to and from the fire, and fuel burn and time, when calculating how much endurance we have in our fuel tanks.

    that statement is something ive always wondered NOT a pilot,but ive always wanted to be one,and my instrest has led me to wonder often…are you “balls to the walls” to and from a fire?..but it sounds by Doug’s comment,id have to say no…makes me think about when i drove race cars..unlike nascar drivers on a 2 mile track,those of us you drove tracks under half mile,we have to use the throttle for car control,,that big right rear car turn the car with throttl feathering as much as the steering wheel.i would imagine its much the same when lining up a drop,and with filight simulator,i learned from bud anderson to use the throttles on both engines to line the plane up for a landing…if thats right or wrong..i really dont know..but its something bud told

  3. Dave, there are practical limitations on how fast an aircraft can go to and from a fire, which include the actual aircraft limitations themselves, as well as legal (maximum speed limitations in given airspace, or below 10,000′), in turbulence, or when heavy, and climbing for terrain with a load. In most cases, it’s not necessarily practical, nor desirable to transit at maximum speed, for fuel economy, safety, etc. The distance to the fire also plays a factor.

    Yes, the throttle is constantly in use, as it’s the means of controlling fuel flow to the engine, and consequently torque or thrust. Speed management during the drop is important, and often a challenge during a downhill run; keeping speed in check may be a problem.

    Much of the math is done by computer rather than in our heads; the basic physics formula of rate/time/distance to calculate our speed, or our time to the fire, or the distance travelled at a given rate and time, is straight forward, but often read right off the GPS. GPS has made identifying targets, terrain, alternate airports, fuel calculations, and many other factors of managing flight data, both simple and accurate.

    While algebra figures into the calculation of things like lift and drag, and such calculations deal with coefficients and so forth, from the cockpit we generally look at pitch angle, power, and airspeed. Mostly airspeed. Many of the calculations that we might use for performance are already done, and printed as data in flight manuals. We simply make note of the temperature, our weight, our elevation, then see what the chart tells us for takeoff distance, climb rate, landing distance, and important things such as the amount of runway necessary to reject a takeoff or continue it in an emergency. Someone else has done the math.

    We might convert weight of fuel to gallons; weight is important for fuel, retardant, etc. But we order the fuel (and the retardant) in gallons. We might convert temperature celsius to farenheight. On a long ferry, we might do a mental calculation regarding when to let down from a cruise altitude based on a descent ratio or rate. Some of the aircraft have flight management systems which do most of this for us, and in other aircraft or situations, it’s up to the pilots to figure this out.

    In the airplane we have an altimeter which gives us basic height information relative to sea level, but the drop altitude is most often done visually, as the terrain is variable and usually different on every drop. Retardant needs to be dropped from a height that allows it to stop its forward motion and drift straight down, to preclude covering just one side of the fuel (“shadowing”). This height, in turn, depends on various factors which include wind, slope, coverage level, fuel type, etc. The drop isn’t an exact science, and in fact is more of an art than a numbers game.

  4. One of the mot amazing parts of a helicopter firefighting pilots job is the ability to take a bucket of water that is dangling 100 feet below a helicopter that is flying 60 IAS knots and is 60 feet above the tree tops that are on a 150 feet below that, on a slope with a downhill approach, into a 15 knot upwind flow and using the most sophisticated computer ever invented provide input to a toggle switch which sends an electrical signal to relay to precisely release a mechanical cam, allowing a carefully disbursed flow of wetting agent over a visually acquired but obscured location and have the solution break the fire triangle so firefighters on the ground can continue suppression efforts, while all the while cyclically and collectively manipulating mechanical/hydraulic servos and communicating at the speed of sound. The human brain is an amazing instrument.

  5. Real world reasons for Algebra are made up by Algebra teachers to keep them in a job…lol

    I haven’t used the Quadratic formula since Algebra class and I have a whole list of certificates, degrees and licenses on top of a helicopter license…

  6. I always thought it was interesting on how a crop duster pilot can turn on and off the seed, liquid or fertilizer usually within ten feet of the fields boarder, consistently? In fire fighting you add the slope, wind and airspeed management; what process does the human brain go through? As for the quadratic formula; maybe trying to figure out the tip at a restaurant?

    1. Back in the day, we used “automatic flagmen” to mark our entry and exit points from the field, or the point where we ran out of chemical on a pass. It was a box on the wing that contained a couple hundred cardboard squares, each with a long crepe streamer attached. To mark the spot, we’d press a trigger button, and a solenoid would kick out one square. The streamer was visible from the cockpit, and it would mark a spot on the ground for us. It was biodegradable and disappeared over time. It didn’t take much practice to be able to lay the streamer over a fence or put it exactly where we wanted, to make it more visible.

      Dropping retardant is similar; it’s a matter of developing a feel for where the retardant will go when the button is pressed. That, in turn, depends on altitude, winds, coverage level, etc. At the beginning of every season, one’s retardant-fu, the intuitive skillset of where to drop the retardant, is low, or weak. It develops as the season rolls on, as one should get better and better at targeting the retardant as one gets more current. Dropping retardant, like most things in aviation, is a perishable skill. The only way to really keep it up or sharpen it is to keep dropping the retardant on fires. Because every drop is different, it’s an evolving skillset.

      I took some online classes a few years ago, including a math class. The instructor kept admonishing the students about the usefulness of the material; learn it well, class, you’ll be using it for the rest of your life. Some of the students were working adults; we agreed that while learning is never a bad thing, most of what was learned had no real application in our day jobs. While there’s a lot of math to be used in some aspects of aviation, it’s more among the engineers than among mechanics and pilots. I use simple math in the cockpit, and fairly simple math as an aircraft mechanic, too.

      When my kids had math homework, sadly I couldn’t do much to help them, as most of it went right over my head.

      1. Doug,thank you for all of your input.Joesph,Adam and Johnny as well,all food for thought,ive sorta of stolen this thread for myself,but very nice info for a flight (sim?) geek like myself.,Doug may i ask where you are located?..what you fly for an Air Tanker?..multi-types?..or S2T?..i have a BAe-146 dressed up in flight yes…im 53 and a little over a month to my 14th birthday…wheeeee…lol

        to the helicopter pilots,all of has always blown my mind to see a pilot leaning into the bubble window,looking straight down and being in full control of the flying machine..i think thats just awesome!takes alot of coordination to fly one of those anyway..then to be twisted to the side,looking straight down,or even down and behind him or her,all the while still being in full control of the aircraft..

  7. Florida Student,

    Here is an example, I hope it helps.

    D = V/4 * H^1/2

    The quadratic equation is the equation for a parabola, which is the ballistic path that a dense object takes in the air (for example, a cannonball in the last few seconds of its flight, after it’s slowed down). It neglects air resistance and so for a water drop I think it would be too inaccurate, over and above the fact that you wouldn’t know the values of the three constants accurately, over and above the fact that it would be very hard to solve while flying.


    Drop distance in feet is aircraft speed (in feet per second) multiplied the square root of height (above the target), all divided by four.

    So if you are flying at 60 mph = 88 feet per second (roughly) and your height above the fire (which is on the ground) is 256 feet, then you would drop 352 feet from the fire: 88/4 = 22, square root 256 = 16, 16 x 22 = 352 feet.

    It’s a simple quadratic equation, but you would still need to do a square root in your head and estimate your height above the fire (not the treetops below you) and estimate the horizontal distance to the fire correctly right at the moment when you release the water. There is no constant sighting angle at which you’d drop the water, and wind would have a non-linear effect which you couldn’t wrap a quadratic equation around.

    In simple terms, it pretty much sums up why a KMAX drone is NEVER going to be able to compute all the variables to be able to support firefighters accurately with water drops.


    Note that if airspeed = zero (a hover) then D = 0, which is the case for a helicopter hovering directly over a fire and this reduces the variables.

    So, a KMAX drone filling Klump pump tanks or heliwells, or making spot drops down a stovepipe might possibly be doable, assuming there is no wind and there is no slope and the bucket isn’t oscillating and it doesn’t need to spend all day getting lined up producing downwash and … Oh, forget it.

    Give me a pilot any day.

  8. Like Doug stated dropping retardant is a perishable skill, unless your Bill Waldman. I started flying ag in the ’60’s. I failed algebra in high school, too many distractions. As mentioned by the other commenters its a feel and site profession. The cockpit is your office. You are a system manager, your life depends on it. Going back to perishable skill; I always wondered how at the beginning of rice season in California pilots would land. load (one minute) and take off on a rice dike one half mile long and fifteen feet wide about 100 times a day. At first you are very aware of the other ag company with loaders (fuel truck) trailers full of product at the opposite end of the dike (1/2 mile). As the spring winds start to blow mid day in the beginning you have to land over the top of the other ag company to avoid landing downwind. I noticed about day three (12 hour day) 300 t&l it really didn’t matter about the wind. Quartering tail wind gusting to 12 mph bi-wing airplane, just keep on working. Somewhere in this grueling work ethic your brain and control responses take over. Your body is just along for the ride. Absorb all the education you can!

  9. As a USAF combat crewmember (rated nav, in B-52s and B-1s) (retired long since), let me say that I stand in awe of the wildland firefighting pilots and what they do. No one who takes wing exhibits greater courage, skill, concern for the public, or devotion to their work.

    In summer 2000, I chanced to hear a caller to C-SPAN’s Washington Journal claim he was a retired USAF fighter pilot. Aviation resources were hard-pressed at the time and he wanted to help.

    He knew there were many F-16s standing idle (this was before 9/11, remember), plus large stocks of jettisonable external fuel tanks. He suggested the wildland firefighters call on those thumb-twiddling fighter pilots for help, fill the unused tanks with retardant, and send the fighters out to drop the tanks on fires.

    I am pretty well-versed in fighter operations, but not so well-versed in what wildland firefighters do (on the ground or above it). But his idea sounded terrible.

    Can anyone enlighten us?

    1. Dean, there have been a number of different ideas on fighting fire from the air, from dropping detergent and glass spheres full of water, to tanks with glass ends that were exploded in flight over the fire to release the water. There was even a proposal to make a manmade raincloud of linked balloons that dropped mass quantities of water.

      Drop tanks full of water, released from F-16’s would be dangerous, I suspect, given both speed and the kinetic energy of a metal or carbon tank. Generally with water, foam, or retardant, we look for the material to be released in the air and to stop its forward motion before reaching the ground. Heavy coverage levels released at once at low altitude can do a lot of damage, even dig ruts in the ground.

      There was a concept suggested several years ago using the A-10 for fire work, but it was never seriously considered. A group of A-10 pilots were trying to create a market for themselves after military life, and pushed the A-10 as a potential platform.

      There are former military pilots flying tankers, in helicopters, single engine (SEATs), and large air tankers. There are former military aircraft flying fires, including helicopters and large air tankers. It might be a stretch to put fast tactical aircraft on the line as air tankers, however, for a number of reasons. Military pilots who are looking for tanker work would probably be best served applying to one of the tanker companies for a pilot position.

  10. In today’s digital world, the quadratic equation resides in every digital navigation computer in an aircraft, including hand held GPS nav systems. It is used by programs that determine the distance between any two navigation coordinates or points. Before computers, we measured the distance between two nav points using a paper chart and a clear plastic scale. Now, we can just input the nav coordinates and the computer figures out the distance using a program that employs the quadratic equation.

    A lot of our flight performance tables we use for flight planning were made by using the quadratic equation. This gives us a shortcut so we don’t have to do quadratic equations every time we go flying.

    Also, you can’t design an airplane without using the quadratic equation for stress and strain analysis.

  11. It has been said after an aviation mishap -“if the pilot survives the crash, you’ll never really know what happened.”

    The air tanker pilot community does not suffer from an excess of washed up financial strategists, computer scientists or even former high school physics teachers. The contrary is true. Most work only 3-8 months per year and spend the rest of the time.. sipping mimosas on a beach in a third world country, commenting on articles Bill publishes through his sites or teaching other potential air tanker drivers how to synthesize all the inputs into successful flight strategies where you rarely die.

    However, air tanker company owners however are very well versed in using the quadratic equations (mostly calculus) to solve for profit (x) using the variables of risk, cost, taxes, pilots, mechanics, contracts, and inefficiency.

    One reason why the GAO is requiring the AFUE study is because the firefighting community has relied on “gut feelings” forever to determine the effectiveness of a firefighting tool relative to its cost. It turns out the taxpayers have a hard time swallowing the idea that we spend hundreds of millions of dollars on aviation to fight fire but cannot tell them how useful (x) a VLAT is vs. 5 SEATs given time of day, crews on site, wind conditions, topography and complementing resources.

    If you learn math, you can figure out how much money you are making per hour compared to another job. Be a student of math and you can write your own future using as many ‘$’ signs as you want.

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