Question:
When landing an aircraft, the pilot needs to calculate the landing distance over a fifty foot (50') object. This calculation is based on headwind speed, weight of the loaded aircraft, altitude and the temperature.The headwind is assumed to be calm.
Example I: If the runway is at 1000 feet altitude and the temperature is 80 degrees and the weight of the loaded aircraft is 4400 pounds; the landing rollout would be roughly 1425 feet.
Example II: If the runway is at 2000 feet altitude and the temperature is 80 degress and the weight of the loaded aircraft is 4400 pounds; the landing rollout would be roughly 1700 feet.
Example III: If the runway is at 2000 feet altitude and the temperature is 60 degrees and the weight of the loaded aircraft is 4400 pounds; the landing rollout would be roughly 1650 feet.
Write a formula to compute the approximate landing distance given the examples.
My Attempt:
I see that the landing distance depends on weight of the aircraft, altitude, and temperature.I then wrote this with function notation: $$f(x,y,z)=D$$ where $x=altitude$ $y=temperature$ $z=weight$ and $D=landing distance$. I then wrote the following examples in function notation. Example 1: $$f(1000,80,4400)=1425$$ Example 2: $$f(2000,80,4400)=1700$$ Example 3: $$f(2000,60,4400)=1650$$ I notice that some of the parameters in each of the examples are the same. I think that means that I can get rid of some variables but I am not sure. I also noticed that as the $x$ and $y$ parameters increase, $D$ increases. I also thought about subtracting the examples from each other, maybe that would eliminate some variables,but I do not know how to justify that because I have no idea if the formula is linear, quadratic, etc.
I tried to look up a formula for landing distance. I suspect the actual formula is a differential equation, but all I could find is a FAA chart for calculating it. An example of how it is calculated can be found here: https://www.youtube.com/watch?v=i4zwzOS2dH4 I wondered if I could use a chart to write a piece-wise function, but I think that would be a step in the wrong direction.
If you need any more information please tell me.
The resulting equation should apparently should be a function of two variables $f(x,y)$.
Any help would be greatly appreciated.
Given only those examples as input, you can make no deductions regarding how the weight of the aircraft affects the landing distance, as the weight is the same in all examples. Your formula can therefore only depend on the altitude ($x$) and the temperature ($y$). Given no further information let's assume that the landing distance $f(x,y)$ is a linear equation, i.e. that: $$f(x,y) = Ax + By + C$$
We then have $3$ equations with $3$ unknowns. Inserting the values and solving, we find that $$f(x,y) = 0.275x+2.5y+950$$
This equation gives the correct landing distances for the $3$ examples, but whether the assumption of linearity is a fair assumption, I don't know.