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Had our choice for a topic for a project, and I picked math modeling.

My Original Research Question: In what way can a playing card be thrown in order to maximize the distance it travels?

My Simplified Research Question: In what way can a flat disc be thrown in order to maximize the distance it travels?

(My original question came from a phenomenon I observed after throwing some playing cards. The card flew in a wide corkscrew motion while traveling forward, and I wondered why, and I tried to do some research and produce a model. However, after finding about all the factors that will be involved, I decided to simplify it in order to not kill myself. I am a high school senior, with an able grasp of mathematics. I have done calculus and am in the process of learning PDE.)

I currently only have two questions.

I have decided to aim for a solution of the POSITION of the center point of the disc, and a normal vector to the disc. It seems to be like this would be the easiest thing to solve for in the end, and easiest thing to model. I also want this model to be able to handle any variable change I throw at it. In order to create this model, I need a list of things that effect the flight.

List

Weight - (Drag)

"Frontal Area"

Surface Area (Drag)

Initial Rotational Velocity (Stability?)

Initial Velocity (Drag and Lift)

Initial Yaw, Pitch, and Roll (Drag and Lift)

Magnus Effect

Air Density (Magnus/ Lift / Drag)

Air Velocity (Magnus/ Lift / Drag)

I know these are a lot of factors, but I want to account for everything. And anyways, without wind, it's a simple projectile problem.

I ran across Reynolds number (a ratio of some sort), Lift coefficient, and Drag coefficient. The problem is, all of these things seemed experimentally determined, which obviously defeats the purpose of my model. Is there anyway to calculate them?

Second, my method of solving for the position would be like this.

1. Find Net Force acting upon disc.

2. Convert to acceleration.

3. Somehow convert to a velocity (I'm thinking a derivative).

4. Somehow convert to a position (I'm thinking a derivative).

I can think of several errors with my reasoning.

1. Different forces are acting upon different parts of the disc, which is what causes the disc to change its orientation.

2. My converting to velocity, I can only think of one equation at the time

v = a t

But, again, this is assuming the disc will not tumble around, or anything of the sort.

So my second question is, how would I do this?

PLEASE NOTE: I am not looking to be fed answers. I know this is graduate level fluid dynamics, and am willing to buy books for whatever knowledge I need. However, I am only a senior at high school, so ANY simplification possible would be GREATLY APPRECIATED. Also, I know this is kind of long, but I just wanted to throw everything I know out there. Currently, all I have is many equations without knowing how to combine them, and that's where PDE come in. I have experience with MATLAB and Mathematica, and am planning for the final model to be inputted to one of those programs (probably MATLAB, because I'm told it is much better for vectors).

I might as well throw in a third question. I am finding conflict between Bernoulli's Theorem and the Coanda effect. In my findings, it is not simply one or the other, but actually a combination of both that accounts for lift/drag. How can I account for this?

UPDATE: I have done some more research and found that

when gravity interacts with the rotational force it produces a torque, which will interfere with the path of flight.

.there will be an interaction between the ball spin angular momentum and the differential drag forces on the ball.

I can honestly say that I have no idea how to input this to my model. It seems to me that all of these forces are acting upon separate points on the disc. I do not know how to account for the magnus effect on a disc, because all I have found online is spherical shaped calculations. For my essay, I will talk about all of these factors, and for the model I will decide what factors to include or not.

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# Aerodynamics: Mathematically Modeling the Flight of an Object

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