Physics? Setting up PDE for air resistance at high velocity

[Mr. Fizzix]

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 realistic-possible 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 first-order, linear, non-homogenous, 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.

 

[fresh_42]

You basically start with $\dfrac{\partial}{\partial t}v = a + k_A\cdot v$ which means for the units $ms^{-2} = [k_A] \cdot ms^{-1}$ and your constant air resistance seems to be time dependent: $[k_A]=\dfrac{1}{s}$. I cannot see, how a constant factor per time unit should make and physical sense.

Not to mention, that in your time independent solution you have $v^2 = \dfrac{a}{k_A}$, i.e. $[k_A]=\dfrac{1}{m}$ and all of a sudden your constant factor seem to depend on distance in a way that velocity tends to zero for large distances and to infinity for short ones, which again doesn't make any sense.

I'm afraid, that your system has some fundamental flaws. Basically $v'(t) = a+c\cdot v(t)$ is easy to solve: http://www.wolframalpha.com/input/?i=v'(t)+=+a+c\cdot+v(t) . However, a time independent acceleration makes me wonder.

 

[Mr. Fizzix]

You basically start with $\dfrac{\partial}{\partial t}v = a + k_A\cdot v$ which means for the units $ms^{-2} = [k_A] \cdot ms^{-1}$ and your constant air resistance seems to be time dependent: $[k_A]=\dfrac{1}{s}$. I cannot see, how a constant factor per time unit should make and physical sense.

Not to mention, that in your time independent solution you have $v^2 = \dfrac{a}{k_A}$, i.e. $[k_A]=\dfrac{1}{m}$ and all of a sudden your constant factor seem to depend on distance in a way that velocity tends to zero for large distances and to infinity for short ones, which again doesn't make any sense.

I'm afraid, that your system has some fundamental flaws. Basically $v'(t) = a+c\cdot v(t)$ is easy to solve: http://www.wolframalpha.com/input/?i=v'(t)+=+a+c\cdot+v(t) . However, a time independent acceleration makes me wonder.

I am not sure how much we are on the same page here. Regarding the "time independent acceleration", I am saying there are different sources of acceleration. We have gravity, g, which indeed is time independent. It is a constant. But also constant is the thrust put out by the engines. Let's say I am running my engines at 80%, and never change that. That term will be a constant, independent of time, the only reason it would be a function of time is if I start getting into response times. But, it can absolutely be treated as a constant for the current scope of this project. Constant sources of acceleration do not necessarily mean that the velocity and position functions will not be functions of time, as you are well aware with falling bodies.

Keep in mind that, as it is a video game, the initial conditions will update, "re-setting" the system every few frames, so as to allow the player to control the aircraft.

I honestly haven't been focusing on units as of yet, as the math has been a struggle so far. Although, you may be onto something.

I did find an article that talks about solving for air resistance that is proportional to the square of velocity, which outlines the process here. Although I am not following why it is using terminal velocity as part of the equation.
https://prettygoodphysics.wikispace...tions.pdf/273523270/DifferentialEquations.pdf

Thank you.