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Not sure if this is more appropriate for physics or for differential equations, but this problem centers around an airplane traveling with air resistance. I am not looking to get the most realisticpossible model, as it is just for a video game I am making.
Although this is technically a PDE, only one independent variable needs to be treated as such; all the others can be treated as constants. So, for all intents and purposes, this can be treated as an ODE.
At first, I decided to use the following article to model air resistance:
http://oregonstate.edu/instruct/mth252h/Bogley/w02/resist.html
That author of the above article solves the problem for "Linear Air Resistance", which I was able to solve for. After several, several steps of doing vector balances, I came up with the following differential equation. I had modified the differential equation from the article above in that, I also included a term for acceleration due to engine thrust. Also, rather than working with forces, I decided to work with accelerations, dividing both sides by the mass (of the aircraft).
I came up with the following equation. I apologize for the crazy nomenclature. Please private message me first if you have any questions regarding that. I could send you the entire file of derivations if you were interested.
Now, as you can see, the acceleration, due to engine thrust, is only a function of throttle setting and not time. This means changes in the throttle setting will mean instantaneous changes in acceleration. This is a simplification of mine that I may later go back and change but, for the scope of this thread, I am not concerned about that.
This is a firstorder, linear, nonhomogenous, partial PDE with constant coefficients. Anyway, I was able to solve this differential equation and get the same result as the author of the above link had, as shown below:
I was also able to prove the solution by plugging it back into the original differential equation.
So, I do feel confident in the math thus far; however, since I am modelling air resistance for bodies at high velocities (near, at, or beyond mach 1), it is said in the link above that air resistance actually proportional to the square of speed, although it doesn't actually show the process.
Nonetheless, I have set up such a differential equation, as shown below:
My main question is: is this PDE set up properly? Is this how you model air resistance squarely proportional to velocity?
Anyway, I have solve the above differential equation by first, using the Riccati equation to turn it into a 2nd order, linear, PDE. From there, I was able to solve it and obtain the following solution:
I checked it against the original PDE, and it is indeed a solution, but something is very wrong. The time parameter was eliminated. No initial conditions, or anything.
This tells me that the PDE was not properly set up. I understand that k_{A} is a simplification to the drag coefficiant. Any idea what may have gone wrong? I would be happy to elaborate upon my process and conventions. Thank you in advance.
Although this is technically a PDE, only one independent variable needs to be treated as such; all the others can be treated as constants. So, for all intents and purposes, this can be treated as an ODE.
At first, I decided to use the following article to model air resistance:
http://oregonstate.edu/instruct/mth252h/Bogley/w02/resist.html
That author of the above article solves the problem for "Linear Air Resistance", which I was able to solve for. After several, several steps of doing vector balances, I came up with the following differential equation. I had modified the differential equation from the article above in that, I also included a term for acceleration due to engine thrust. Also, rather than working with forces, I decided to work with accelerations, dividing both sides by the mass (of the aircraft).
I came up with the following equation. I apologize for the crazy nomenclature. Please private message me first if you have any questions regarding that. I could send you the entire file of derivations if you were interested.
Now, as you can see, the acceleration, due to engine thrust, is only a function of throttle setting and not time. This means changes in the throttle setting will mean instantaneous changes in acceleration. This is a simplification of mine that I may later go back and change but, for the scope of this thread, I am not concerned about that.
This is a firstorder, linear, nonhomogenous, partial PDE with constant coefficients. Anyway, I was able to solve this differential equation and get the same result as the author of the above link had, as shown below:
I was also able to prove the solution by plugging it back into the original differential equation.
So, I do feel confident in the math thus far; however, since I am modelling air resistance for bodies at high velocities (near, at, or beyond mach 1), it is said in the link above that air resistance actually proportional to the square of speed, although it doesn't actually show the process.
Nonetheless, I have set up such a differential equation, as shown below:
My main question is: is this PDE set up properly? Is this how you model air resistance squarely proportional to velocity?
Anyway, I have solve the above differential equation by first, using the Riccati equation to turn it into a 2nd order, linear, PDE. From there, I was able to solve it and obtain the following solution:
I checked it against the original PDE, and it is indeed a solution, but something is very wrong. The time parameter was eliminated. No initial conditions, or anything.
This tells me that the PDE was not properly set up. I understand that k_{A} is a simplification to the drag coefficiant. Any idea what may have gone wrong? I would be happy to elaborate upon my process and conventions. Thank you in advance.
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