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Projectile motion of tennis ball with air resistance

  1. Dec 28, 2013 #1
    Hi, so I am trying to model the path of a tennis ball when serving. I already have the model without air resistance, but now I'm getting into differential equations with the air resistance. I obtained two differential equations for acceleration that i think are correct, but I'm not sure where to go from here exactly. It is in two dimensions. What I want to find is what angle the ball must be hit at to travel a set distance.


    So far these are what i have:
    equation 1: m[itex]\stackrel{d^{2}x}{dt^{2}}[/itex] = -k [itex]\stackrel{dx}{dt}[/itex] [itex]\sqrt{\stackrel{dx}{dt}^{2}+\stackrel{dy}{dt}^{2}}[/itex]

    equation 2: m[itex]\stackrel{d^{2}y}{dt^{2}}[/itex] = k [itex]\stackrel{dy}{dt}[/itex] [itex]\sqrt{\stackrel{dx}{dt}^{2}+\stackrel{dy}{dt}^{2}}[/itex] - g

    K is drag co-efficient, m is mass, g is gravity acceleration, all of which are defined.I have the initial velocity, and I know that I can use trig ratios to get the initial x and y velocities, but I have not idea what to do with them. BTW all the above below things are meant to be fractions, but i am not sure how to make them fractions, sorry.
     
  2. jcsd
  3. Dec 28, 2013 #2

    haruspex

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    Assuming drag is proportional to the square of the speed (which is a fair approximation at high speeds) then your equations are basically right. You do have the sign wrong on the drag term in the y-equation. Drag will always oppose the direction of movement.
    For the LaTex, put the m inside the LaTex, use \frac, not \stackrel, and wrap the dx/dt terms in braces so that the squaring applies to the whole term, not just the x or y:
    [itex]m\frac{d^{2}x}{dt^{2}}[/itex] = -k [itex]\frac{dx}{dt}[/itex] [itex]\sqrt{{\frac{dx}{dt}}^{2}+{\frac{dy}{dt}}^{2}}[/itex]
    As to solutions, I believe there is no closed form solution.
     
  4. Dec 29, 2013 #3

    ehild

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  5. Dec 29, 2013 #4
    Thank; I do know that there is no closed solution but I am wondering how to do the mass calculations on something like excel? I am not sure how to incorporate these two equations for acceleration in my position equation, or in my calculations for range. Also, I think my sign is correct since the ball is being hit downwards, which I've made negative. I think i have to integrate the equations, but I'm not sure in what way I need to use them yet.
     
    Last edited: Dec 29, 2013
  6. Dec 29, 2013 #5

    ehild

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    Last edited: Dec 29, 2013
  7. Dec 30, 2013 #6

    haruspex

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    The contribution of drag to the acceleration must be a negative multiple of the speed.
     
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