There’s a lot of math to flying airplanes, quite a lot of it you’d recognize from high school physics class.
Pilots are required to calculate the weight and balance of the aircraft before flight, making sure the plane is not only below maximum gross weight so that the airplane can safely carry the load, but that the center of gravity falls within an acceptable envelope so that it remains controllable… This is done by weighing everything including the airplane itself, and noting their location from a datum point. Multiply the weight of each item by their location aft (or sometimes forward but the datum is usually somewhere near the front of the plane) and you get a “moment arm” of that item. Add the weights and moments up, then divide the total moment by the total weight to find the location of the CG. Weight * Arm = Moment.
Often you do this with everything but fuel to calculate how much fuel in pounds you can carry given your intended load of passengers and baggage. Avgas is usually handled in gallons not pounds, so convert from weight to volume. Then look up your fuel consumption rate for not only the cruise portion of flight but startup, taxi, takeoff and climb to altitude. That will tell you your endurance plus the legally mandated 30 minute reserve (45 minutes at night). Determine how long you can be in the air without running out of fuel.
From there, you’ll need to plot your course on a sectional chart, measure the true courses, then correct for magnetic variation and compass deviation, then get the winds aloft forecast and do the trigonometry to find the wind correction angle to calculate your compass heading and your ground speed, and then given the distance of each leg and your ground speed during those legs you can compute your time aloft. This information will be used on the ground to make sure you can make the intended flight within your earlier calculated endurance, “will we make it without running out of fuel,” plus you will turn to the computed compass heading to not get blown off course and get lost.
Then there’s my favorite aviation equation, I found this in the Pilot’s Handbook of Aeronautical Knowledge: An airplane tire will hydroplane at a speed in knots equal to the square root of the tire pressure in PSI. The reason the government published that equation was so pilots could calculate whether they can use full braking for short field landings if there is standing water on the runway. For example, the main tires of a Cessna 172 are inflated to 35 PSI, let’s round that to 36 to make the math easy, the square root of 36 is 6, 6 times 9 is 54, a Cessna 172 will hydroplane at 54 knots. The VS0 of this plane is 44 knots, a touchdown speed of 1.2VS0 comes out to be about 53 knots, so if you manage your energy well on approach you will just barely have full braking available immediately after touchdown even with standing water on the runway. There’s a more fun off-label use for this info though: if you have fairly low pressure tires, like big balloon tundra tires on a bush plane, you can set down on a lake and hydroplane well below your VS0 then just water ski up to the shore, set the tail down and stop to get in and out of somewhere where you don’t have room to land on solid ground.
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There is a question that high school algebra teachers get asked a lot that I never heard: “Why do we need to learn this?” When I teach the above math, I answer that question myself. I did so in the paragraphs above, I explained why it is important for pilots do to these things. This is an application of the Principle of Readiness as described by Ed Thorndike, which…think about Maslowe’s hierarchy of needs. I’m teaching flight school, you’re teaching high school algebra. Where do our lessons fall on our students’ pyramid of needs? Probably pretty high up, right? Somewhere in the Self Actualization layer? So first of all, you need to make sure all the students’ needs that are below that on the pyramid are taken care of so that they’re operating at the level of your lessons, and once you achieve that, you MUST establish why and how your lesson will fulfill their needs.
As a flight instructor, that answer almost always boils down to “so you don’t crash the plane and get hurt or killed.” Algebra, geometry, trigonometry and calculus teachers are THE LITERAL WORST at this, coming up with answers like “This might save your life someday” or “it’s required to get your diploma.”
The former answer is so vague that it tastes like a lie; it might as well be “just in case you’re ever on Jeopardy and the Daily Double is about this topic.” The latter is an outright insult. K-12 students are required by law to go to school, and by the time they’re in Algebra class they’re old enough to understand the social contract allegedly at play here. “All these classes are supposed to prepare me for life as a citizen of this society, to equip me with the knowledge and skill I will need as a functioning adult responsible for earning my own living.” And then they drop “We have unilaterally decided to waste your time by requiring that you ‘learn’ something that I as the subject matter expert acknowledge is fully useless to you. We have officially departed the ‘preparing you for life’ portion of your education and have now entered the ‘how much bullshit can you swallow’ portion. Test is on Friday.”
I partially blame the approach taken to building mathematics curricula. We start out teaching pre-algebra to 13 year olds as if they’re going to grow up to be mathematicians. You’re trying to hit hormonal pubertines with shit like “the transitive property of inequality” and expect them to internalize it for dumbass reasons like “someone decided it’s mandatory I guess.”
This attitude of vindictive joy at proving your old math teacher wrong, “See? I never use this!”…I can only explain it as an abject failure on the part of the teacher/school/state department of education for designing and teaching the curriculum the way they do; not only failing to motivate students to learn, but building in them a pride in their resulting ignorance. Short of physically injuring your students I don’t know how you fuck up worse than that as a teacher.
There’s a lot of math to flying airplanes, quite a lot of it you’d recognize from high school physics class.
Pilots are required to calculate the weight and balance of the aircraft before flight, making sure the plane is not only below maximum gross weight so that the airplane can safely carry the load, but that the center of gravity falls within an acceptable envelope so that it remains controllable… This is done by weighing everything including the airplane itself, and noting their location from a datum point. Multiply the weight of each item by their location aft (or sometimes forward but the datum is usually somewhere near the front of the plane) and you get a “moment arm” of that item. Add the weights and moments up, then divide the total moment by the total weight to find the location of the CG. Weight * Arm = Moment.
Often you do this with everything but fuel to calculate how much fuel in pounds you can carry given your intended load of passengers and baggage. Avgas is usually handled in gallons not pounds, so convert from weight to volume. Then look up your fuel consumption rate for not only the cruise portion of flight but startup, taxi, takeoff and climb to altitude. That will tell you your endurance plus the legally mandated 30 minute reserve (45 minutes at night). Determine how long you can be in the air without running out of fuel.
From there, you’ll need to plot your course on a sectional chart, measure the true courses, then correct for magnetic variation and compass deviation, then get the winds aloft forecast and do the trigonometry to find the wind correction angle to calculate your compass heading and your ground speed, and then given the distance of each leg and your ground speed during those legs you can compute your time aloft. This information will be used on the ground to make sure you can make the intended flight within your earlier calculated endurance, “will we make it without running out of fuel,” plus you will turn to the computed compass heading to not get blown off course and get lost.
Then there’s my favorite aviation equation, I found this in the Pilot’s Handbook of Aeronautical Knowledge: An airplane tire will hydroplane at a speed in knots equal to the square root of the tire pressure in PSI. The reason the government published that equation was so pilots could calculate whether they can use full braking for short field landings if there is standing water on the runway. For example, the main tires of a Cessna 172 are inflated to 35 PSI, let’s round that to 36 to make the math easy, the square root of 36 is 6, 6 times 9 is 54, a Cessna 172 will hydroplane at 54 knots. The VS0 of this plane is 44 knots, a touchdown speed of 1.2VS0 comes out to be about 53 knots, so if you manage your energy well on approach you will just barely have full braking available immediately after touchdown even with standing water on the runway. There’s a more fun off-label use for this info though: if you have fairly low pressure tires, like big balloon tundra tires on a bush plane, you can set down on a lake and hydroplane well below your VS0 then just water ski up to the shore, set the tail down and stop to get in and out of somewhere where you don’t have room to land on solid ground.
=====
There is a question that high school algebra teachers get asked a lot that I never heard: “Why do we need to learn this?” When I teach the above math, I answer that question myself. I did so in the paragraphs above, I explained why it is important for pilots do to these things. This is an application of the Principle of Readiness as described by Ed Thorndike, which…think about Maslowe’s hierarchy of needs. I’m teaching flight school, you’re teaching high school algebra. Where do our lessons fall on our students’ pyramid of needs? Probably pretty high up, right? Somewhere in the Self Actualization layer? So first of all, you need to make sure all the students’ needs that are below that on the pyramid are taken care of so that they’re operating at the level of your lessons, and once you achieve that, you MUST establish why and how your lesson will fulfill their needs.
As a flight instructor, that answer almost always boils down to “so you don’t crash the plane and get hurt or killed.” Algebra, geometry, trigonometry and calculus teachers are THE LITERAL WORST at this, coming up with answers like “This might save your life someday” or “it’s required to get your diploma.”
The former answer is so vague that it tastes like a lie; it might as well be “just in case you’re ever on Jeopardy and the Daily Double is about this topic.” The latter is an outright insult. K-12 students are required by law to go to school, and by the time they’re in Algebra class they’re old enough to understand the social contract allegedly at play here. “All these classes are supposed to prepare me for life as a citizen of this society, to equip me with the knowledge and skill I will need as a functioning adult responsible for earning my own living.” And then they drop “We have unilaterally decided to waste your time by requiring that you ‘learn’ something that I as the subject matter expert acknowledge is fully useless to you. We have officially departed the ‘preparing you for life’ portion of your education and have now entered the ‘how much bullshit can you swallow’ portion. Test is on Friday.”
I partially blame the approach taken to building mathematics curricula. We start out teaching pre-algebra to 13 year olds as if they’re going to grow up to be mathematicians. You’re trying to hit hormonal pubertines with shit like “the transitive property of inequality” and expect them to internalize it for dumbass reasons like “someone decided it’s mandatory I guess.”
This attitude of vindictive joy at proving your old math teacher wrong, “See? I never use this!”…I can only explain it as an abject failure on the part of the teacher/school/state department of education for designing and teaching the curriculum the way they do; not only failing to motivate students to learn, but building in them a pride in their resulting ignorance. Short of physically injuring your students I don’t know how you fuck up worse than that as a teacher.
Your high school had physics?! My god man. I couldn’t even pass algebra.