Hey guys. I know I haven't been on in a while and that is just do to school and work, but I need your help. Some of you may know my projectile motion program that I had made (some of you may have even downloaded it!) and read in the readme.txt that that program didn't account for air resistance.

After some time, I found some equations that seemed to work at first, but than didn't when I tested them more closely. These equation I will post at the bottom of this post.

If anyone of you is smart enough to help me find an actual, working equation for the motion of a projectile being shot, accounting for air resistance, I will thank them immensely and if and when I put that program online, I will have you as the author (if I can't do that, I will list you as co-author and in the description say that this is your program).

Here are the two equations:

x(T) = (M/K)Vcos(theta)(1-e^(-KT/M))

y(T) = (-MGT/K)+(M/K)(Vsin(theta)+(MG/K))(1-e^-KT/M))

Variables:
M: Mass
G: Gravity (9.81m/s)
V: Initial Velocity
theta: Angle
K = (MG)/sqrt((2mg)/(CPA))
C: Drag Coefficient
P: Air Density
A: Area of the Projectile

Edit: I must mention that this is for a trebuchet project tau my physics class is doing. We want to try to launch an object into something.

What is wrong with just graphing the equation as a parametric equation?
If you must graph it as a function, convert the parametric equation into a function by solving for T for both x(T) and y(T). Then, set them equal to each other and solve for the y(x). However, this is going to be ugly.
Edit: Actually, this can be easy. Solve both equations for (1-e^(-KT/M)). Then set them equal to each other.

These are parametric equations.

I have only successfully worked out problems where the drag force is linearly proportional to the velocity of the object and I understand how the velocity tends to logarithmically approach some value. However, I tried looking at the more realistic case of the drag force being proportional to the square of the velocity and worked out the integration. I was actually surprised I was able to work it out and get past the integration by recognizing some integrals that were similar to standard ones in an integration table.

In any case, I will link my work out here, and you are more than welcome to test them. If you want the case where we have linearly dependent drag on velocity, that is even easier to work out, and I'd be happy to do so in the event that this is overcomplicated/doesn't work for some reason. (on another note, the equations look "correct" that you have listed-- they are for linearly dependent drag on velocity) I have boxed both the velocities and positions as a function of time in the x and y direction respectively, as well as a coefficient I condensed down, "k", which you will also find boxed, and in terms of familiar things such as the drag coefficient, cross sectional area of the object, etc.

https://dl.dropboxusercontent.com/u/28842208/cemetech/Projectile-1.jpg

Replace the "1" at the end of the file name with 2 or 3 to see the 2nd and 3rd pages.

Please feel free to let me know if you need clarification on anything I did or any of the results. I'm not very familiar with the programming, but I can help with the physics.

Never mind, I got everything that I need, thanks. I don't think that I will update the program, though, because I don't really want to reupload everything. Thanks for all the help again.