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Differential equation for projectile

  1. Sep 3, 2014 #1
    I want to model a Differential Equation for a projectile motion under 2 forces (gravity and wind)

    So, what I have now is an algorithm that simulate the parametric motion (2d) of the project under those 2 forces (given a P position of the projectile with velocity V under a vector of forces F (or acceleration as my projectiles have no mass), find a new position for each time T)
    (Link to the motion simulation example: http://s12.postimg.org/wbr8tyej1/projectile.png)

    I want to model it as a differential equation. If possible with these variants:

    1- basic differential equation projectile motion with no wind, just gravity...the quadratic equation from basic physics should be the solution to it

    2- the "complete" form with wind and gravity as I described

    3- add drag force. I know that the equation may have no solution

    I've learned to solve them but not to model them; if you could give me or point a good text book that show how to model it I'd be glad
    Last edited: Sep 3, 2014
  2. jcsd
  3. Sep 3, 2014 #2


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    So the only force is gravity- since "F= ma" and the force due to gravity is mg, mg= ma, a= g, a constant. The differential equations are [itex]d^2x/dt^2= 0[/itex] since there is no force horizontally and [itex]d^2y/dt^2= -g[/itex] vertically

    These two are a lot the same. The force the wind applies is a form of "drag". The one thing to do is to model it as a quadratic function of the speed, [itex]\omega v^2= \omega (v_x^2+ v_y^2)= \omega((dx/dt)^2+ (dy/dt)^2)[/itex]. That would give [itex]d^2x/dt^2= -g- \omega ((dx/dt)^2+ (dy/dt)^2)[/itex], [itex]d^2y/dt^2= -g- \omega ((dx/dt)^2+ (dy/dt)^2)[/itex].

  4. Sep 3, 2014 #3
    Using your example I was able to see how to reach the first example (only gravity) and found the quadratic equation as solution

    However I had no time to test the second example (gravity+wind) in fact I did not get the general idea, maybe I'm not seeing something but here is what I've found based on my simulation algorithm and your first example

    [tex]y(t)=y_i + v*t*sin(\beta) - \frac{1}{2}y''(t)*t^2 + w*t^2*sin(\alpha)[/tex]
    [tex]x(t)=x_i + v*t*cos(\beta) + w*t^2*cos(\alpha)[/tex]
    position_y = velocity*time - gravity*t^2 + wind*t^2
    position_x = velocity*time + wind*t^2

    x(t) and y(t) are the projectile position at any given time t
    v is the initial velocity of the projectile at launch
    β is the launch angle
    α is the wind acceleration angle
    w is the wind acceleration
    xi, yi are the initial positions on a 2d space

    This lead me to some other questions:
    -Since its me who gives the launch speed (known values), my approach to use them as v and angle β instead of x'(t) and y'(t)? I'm assuming this approach will remove the initial value problem

    -The general idea, seems to be correct?
    Last edited: Sep 3, 2014
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