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Projectile Motion from a height with Air Resistance

  1. Sep 25, 2012 #1
    Hey all, me again. This time my question has to do with projectile motion with air resistance from a given height.
    1. The problem statement, all variables and given/known data
    A cannon is located on a cliff of height h. If the muzzle velocity of a projectile is v0, find the range of the projectile when the drag is proportional to the velocity.


    2. Relevant equations
    ma=-mg-kvy. Vertical motion
    ma=-kvx. Horizontal motion
    mg=kvt. Terminal velocity


    3. The attempt at a solution
    I first attempted to find t via the horizontal motion formula, but my answer doesn't take into account the height difference. Then I tried integrating the vertical motion equation and got an answer of
    t=-vt/g * ln((vt+vy)/(vt+voy)) where voy is the initial vertical velocity. I then made vy=dy/dt and tried to find t from there but I have no idea how to do the integration.

    Am I even on the right path? I'm hoping if I can find the time t given y I can plug that into xcosθ*t to give me the range.
     
  2. jcsd
  3. Sep 25, 2012 #2

    rude man

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    Homework Helper
    Gold Member

    ƩFx = md2x/dt2 → first 2nd-order linear diff. eq. in x
    ƩFy = md2y/dt2 → second linear diff. eq. in y

    You know the initial conditions (4 of them for the two equations) & they're linear constant-coefficient, so solution is straight-forward.

    (I would use Laplace transform but that's 'cause I'm an EE. EE's can't so much as tie their shoelaces
    without transforming to Laplace! :smile:)
     
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