Calculus Aviation Project

I read this:





maybe too simply. Weather might make you choose a different descent rate, an engine fire, a passenger or pilot with a sinus or ear problem, ATC instructions, ...


Didn't even mention to the poor guy that decent rate is usually limited by the speed at which the structure starts bending or breaking. :)

Hey Everyone, Just wanted to say thanks for all the feedback and ideas. This is very helpful and It will give me a lot to research and incorporate in the project. It's greatly appreciated. Honestly, the project was quite open ended so although I think it was mostly about descent paths I'll look at everything and see what else I can bring in.


The assignment as given is so open ended you could spend a lifetime trying to model it. I think the prof simply wants to provide a way to make the math of a triangle interesting by saying it's the "flight path of an airplane".

You could have some fun and ask him if the model is allowed to a) be done for the aircraft when the wing isn't flying at all, say, in an aerodynamic spin or stall... b) be modeled for an aircraft that exceeded its structural limits and ripped the wings off. (Ballistics...)

Heh heh. Honestly we've all taken the question way too far. Have fun and enjoy the math. The real world is messy. ;)
 
Another descent option: regardless of cruise altitude (from 2500-10,000 feet) where I indicate 135-145 mph (higher is faster due to thinner air, reduced drag, etc.; True Airspeed is generally Indicated + 2% per thousand feet of altitude), I descend to 1000' above the airport at a constant 500 feet per minute, and let the airspeed increase. This generally makes the airspeed indicator point to 170-175 mph and groundspeed can be 200 or more. I take the free acceleration to make up for fighting it to reach altitude at slower speed. You can generate a good free body diagram with this, just convert everything to feet per minute first.

Then I level off, slow down from drag (give it 2-3 miles to do so), reduce throttle to slow down to 120 mph or less so I can begin to configure for landing. I generally fly around the airport at 90 mph, descending around 3° and reaching the runway at 70 mph, slowing to stall speed (mid-50s depending on flap position) before touching.

Considerations affecting descent: I start based on time, as 500 fpm is 2 minutes per thousand feet. If field elevation is 567' and I'm flying at 7500, I need to lose 6000' in 12 minutes, plus a couple of minutes to decelerate, so I start down when I am 14 minutes out. Do I start on time, or late? How well do I hold constant 500 fpm? What are the winds at cruise altitude,and how do they change as I descend? Near the surface, winds are generally slower and often shift counterclockwise due to the no-slip condition and Coriolis force from earth's rotation. Call it 15-20 seconds to initiate descent and stabilize rate, and the same to level off and begin slowing.

For a good visualization of winds changing with altitude, check this: https://www.windyty.com/?975h,29.361,-85.485,6

Don't forget that flight path and wind are both vectors, and apwhen level and slow I need to align my flight path parallel to the runway. Wind is rarely in line with the runway. Landing is made into the wind as much as possible.

Then there are things like other planes I may need to avoid during descent; if I am IFR, then ATC may clear me to descend to 6000', where I level off then may be cleared to descend to 4000' or even 3000', and I must level off again and wait to be cleared down to my desired 1600'. Sometimes I will be cleared for further descent before reaching the first clearance and do not level off; I like this!

Hope your project goes well!

For my fellow pilots, your airplane is experiencing acceleration all day; so are you. Gravity is not a constant force, it's an acceleration, 32.2 ft/second squared. Drop a shoe off your bed, then throw the other up near the ceiling. With. I acceleration, their descents would be at the same speed and they would hit the floor with the same force. Does it happen that way? That's how I accelerate making a power on descent, gravity gives it to me free. If I don't pay attention, gravity will pull me right past Vne, all still for free.
 
This is absolutely wrong. It all comes down to calculations involving pee.

:lol:

That is definitely a factor as the inverse square of bladder volume needs to be determined as a function of time remaining before volume remaining approaches a negative number.

:confused:

If that does not make sense to you, please send your comments to Revolution Brewery, Chicago IL. They have had a contributory effect on my diminishing mental capabilities tonight.
 
Hello Pilots, I’m not a pilot although I’d love to be one someday. I’m currently a college student studying engineering. I’m taking Calculus this semester and have been assigned to do a semester project about aviation. Specifically, I’ll be using calculus to model descent paths using polynomials and working with vertical decent rates and acceleration. The assignment prompt talks about three different mathematical conditions that determine approach path, namely cruising altitude, horizontal distance from touchdown and decent rate. The second part of the project asks me to contact a real pilot and find out from them what other factors influence descent from a pilot’s perspective. I was wondering if anyone here would be willing to share what you know about descent from your experience as a pilot. Thanks in advance.

PM sent.
 
Wow you all get trolled easy......funny to watch your responses
 
What if there is an engine/cabin fire
 
I never took calculus, but I can still land the plane. :)

When planning a descent I figure the difference from my cruise altitude to pattern altitude and figure the number of minutes to lose that much altitude at 500fpm. Then I add two minutes so that I don't arrive at pattern altitude directly over the airport. Adjust as needed for terrain, active runway, wind and traffic.

I'll generally start pulling power back 1/2 to 2/3 of the way through the descent, again depending on wind, turbulence and other factors. It also depends how long the descent is since I don't want to shock-cool the engine.
 
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I figure the difference from my cruise altitude to pattern altitude and figure the number of minutes to lose that much altitude at 500fpm. Then I add two minutes so that I don't arrive at pattern altitude directly over the airport. Adjust as needed for terrain, active runway, wind and traffic.

Shouldn't you subtract 2 minutes? :dunno:
 
I think if you did a real world profile evaluation, you'll find the mean profile to be around 600fpm in descent in unpressurized airplanes. This takes into account typical physiological as well as performance issues. You want to keep enough power in to keep the engine(s) properly warm, typically bottom of the green arc as portrayed on the power instruments, yet you don't want to exceed structural speed limitations which vary depending on the turbulence level. Typically unless you are in smooth air, you want to stay around Vno, Velocity normal operating, which is top of the green arc on the airspeed indicator. In very rough air you want to keep your speed to Va, a speed at which you will stall before you can develop enough G loading on the wings to break them. This protects you to a very large degree from structural failure due to turbulence, however there are no absolutes. Va is roughly on the order of 60% or so of Vno. If the air is extremely smooth, now you can hit Vne, Never Exceed (Red line, it's a goal, not a limitation!), beyond which for small planes, one runs into structural flutter potential. Flutter is fatal more often than not. Most GA planes with the power at the bottom of the green arc doing Vno will be descending between 600-1000fpm.

This actually leads to a pretty straight forward possible comparative project, the effect of turbulence on the efficiency of a descent profile.
 
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Frankly, I think the instructor is clueless about aviation and doesn't understand how to realistically define or manage a classroom project. I've been guilty of this more than once (not managing the level of effort) and have had to backtrack on an assignment.
 
Frankly, I think the instructor is clueless about aviation and doesn't understand how to realistically define or manage a classroom project. I've been guilty of this more than once (not managing the level of effort) and have had to backtrack on an assignment.

It could be, it looks like a one or the other. Either that or it's an exercise in understanding that some problems are infinitely complex, and how to handle that. If there is one thing aviation is good for is adding infinite complexity to a problem. Pretty much all of ADM is about handling infinite complexity in real time decision making.
 
I recently did a dead stick landing and the thing I was concerned most about was wasting all the energy I had at 3100 ft to land on the runway in front of me at 600 feet.
For your problem you may want to include fuel consumption to get a precise weight. We pilots just feel it and compensate but you can't do that on paper.
 
Maybe a graph showing the relationships of different flight profiles vs the drop in wallet weight?
 
Not calculus, but rule of thumb; to maintain a 3 degree glidepath you would multiply the altitude to lose by 3 to determine when to start down. If you need to lose 10,000' you start 30 miles out. To determine the rate of descent, it is groundspeed X 5. So if you are going 100 knots you need a 500 fpm descent. Note that these formulas are mostly used in pressurized airplanes as a 200 knot groundspeed would necessitate a 1000 fpm descent, which might not be great for passenger comfort.
 
Not calculus, but rule of thumb; to maintain a 3 degree glidepath you would multiply the altitude to lose by 3 to determine when to start down. If you need to lose 10,000' you start 30 miles out. To determine the rate of descent, it is groundspeed X 5. So if you are going 100 knots you need a 500 fpm descent. Note that these formulas are mostly used in pressurized airplanes as a 200 knot groundspeed would necessitate a 1000 fpm descent, which might not be great for passenger comfort.

Well the OP could take it as a project to confirm that rule of thumb. i.e. figure out the rate of descent needed for a given groundspeed then show how that rate of descent can be approximated as you described above.
 
Shouldn't you subtract 2 minutes? :dunno:

No. Lets say I am returning to Reno (4500') and I am at 9500'. I know I need to lose 5000' at 500fpm. That is 10 minutes. I then add 2 minutes and figure I need to start down 12 minutes before ETA.
 
No. Lets say I am returning to Reno (4500') and I am at 9500'. I know I need to lose 5000' at 500fpm. That is 10 minutes. I then add 2 minutes and figure I need to start down 12 minutes before ETA.

Ah got it.
You are starting 2 min earlier not tacking 2 min on to the end.
Makes sense. I am going to start doing this.

I generally just ballpark it w/o much calculation
 
Ah got it.
You are starting 2 min earlier not tacking 2 min on to the end.
Makes sense. I am going to start doing this.

I generally just ballpark it w/o much calculation

If you have 430Ws you can set one of them up to do the vertical nav calc for you.
 
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