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Aerodynamics: Mathematically Modeling the Flight of an Object

  1. Sep 1, 2007 #1
    1. The problem statement, all variables and given/known data

    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.

    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.
  2. jcsd
  3. Sep 1, 2007 #2
    good question chief, didnt ya ask cortana?? she's a smart AI.
    anyways i too did a lil of research on this, i ll answer u lator. i have to finish somethig important now. till then, you keep the covanent away.. lolzzz
  4. Sep 1, 2007 #3
    er... thanks.
  5. Sep 2, 2007 #4

    i dont know how much understandable i can make my response, but here goes
    ANSWER: high lift, low drag...
    lift= density*velocity^2*area*coefficient of lift/2
    drag= density*velocity^2*area*coefficient of drag/2
    coefficient of lift/drag is proportional to angle of attack and the geometry of the airfoil(a line in case of a card or a flat disk).
    actually Cp = [P - P(freestream)]/[(density*v(freestream)^2)/2]
    and Cl = integral from LE to TE[Cp(lower surface) - Cp(upper surface)]dx
    now the mathematical approach to it is a bit complex.

    for maximizing distance, Cl has to be high and drag has to be low.
    now the rotating part comes in. the rotation of card or the disk helps in maintaing a pitch up configuration(ever heard of gyroscopic couple)
    now to the solution that you want to make.
    the situation is same as that of a ideal projectile motion. but thats IDEAL.
    here the downward force is not constant(weight is, but the lift is varying), neither is the drag zero(as in the ideal case). although both are related to the velocity of the card. so they can be calculated. assume some velocity and you have the drag for it. so you can get the x position. similarly calculate lift and you can calculate y position. of course angle of attack ll vary but assuming it to be constant ll simplify things. for example, assume a valur which has highest lift/drag ratio

    neither the disk or the card is rotating in the vertical plane, so magnus effect can be neglected
    yes there is, but that is far too complex .

    you know i can go on and on and on, but neither you nor me ll gain from it (ok i accept, i ,myself, am a lil bit confused about where to start). i think i have answered a fair amount of your question(it remains to be seen how much you can make out of my response though:redface::redface:)
    suggestion: start now man :wink:. keep posting problems that you encounter.

    anyways plz correct me if i am wrong somewhere
  6. Sep 2, 2007 #5
    Wow. Thanks for all your help. But I think I need to clarify some things.

    First, the magnus effect, in this situation, will not produce 'vertical' lift. Look at :47s and 1:12 of this video, to see what I mean.

    See how the card curves right or left? That is the hardest thing I'm modeling, because it is actually significant to the flight. For my paper, I'm going to discuss all ( as in EVERY SINGLE ONE) the forces acting upon the disc, and for the model, only magnus, basic lift, and basic drag. I think I will just find a Reynolds number, Lift Coefficient, and Drag coefficient online somewhere. It shouldn't be too difficult. With all the equations and coeffecients, I can start to make a crack at this model. I only have to do the magnus effect.

    magnus effect - spinning = unbalanced velocity at each side of the card --> pressure difference --> force exerted on the 'horizontal area' exposed.

    I hope the above is right, and if it is, can i make my own function for the yaw/roll?
    Last edited by a moderator: Sep 25, 2014
  7. Sep 2, 2007 #6
    oh thats wrong i guess, i didnt give it much thought. now that turning sidways stuff is due to the gyroscopic effect. but that is by no means magnus effect. card is too thin to produce such effect
  8. Sep 3, 2007 #7
    oops i am wrong here also, not much though. the plane has nothing to do with the magnus effect. it still can be observed if the disk is rotating in the horizontal plane.
    but this effect is still negligible or nil in our case as the card is too thin to generate appreciable amount of force
  9. Sep 3, 2007 #8
    to add a lil more, that turning effect is observed because the card is not oerfectly horizontal, it is tilted or banked as in a aircraft(rolling), now the lift produced is perpendicular to the plane of card, therefore there is a component of this force perpendicular to the line of motion(thats how an aircraft turns). that is why the turning is observed
  10. Sep 4, 2007 #9
    oho.. so what i thought was magnus is actually gyroscope... ok sweet. I have everything now. Gyroscope, lift equations, and drag equations. Thanks a lot ank_gl, you really helped me get started. I'm very surprised that only one person replied to me in both forums, even with around 120 views. Is that normal for a high level question like this?
  11. Sep 6, 2007 #10
    err.. dont know:frown:
    but all MY stupid question gets answered though:wink:

    one more thing, i only tried. you may wish to take some good advice from your teachers/professors. i am still a student myself
    Last edited: Sep 6, 2007
  12. Sep 6, 2007 #11


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    Staff: Mentor

    Modeling flying cards is non-trivial becuase of the rotation and non-uniform cross-section - rectangular - which means complicated moments. The torques induced on the card are highly variable

    Add to that the low mass and any variation in the local air, and the problem is complicated.

    A disc with a uniform boundary - circle - would be much simpler.
  13. Sep 6, 2007 #12


    User Avatar
    Science Advisor

    I would think that one could also argue a similar effect with a helicopter rotor. Assuming that the card is traveling through the air and since it is pitched slightly, it is generating a small amount of lift. Well since the card is rectangular, as it spins, it "chops" at the air similar to a rotor (although not as nearly since its more square). However, since it is rotating and translating, the side that is spinning with the movement will generate more lift than the opposite side, because it is moving faster through the air.
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