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System of linear differential equation

  1. Apr 12, 2009 #1
    1. The problem statement, all variables and given/known data
    A projectile of mass m is fired at angle theta with the horizontal with initial velocity of v0 (ft/s). neglect all forces except for gravity and air resistance and assume air resistance is equal to k times velocity (ft/s)
    a) with x-horizontal and y-vertical, show that the differential equations of the resulting motion are:
    mx'' + kx' = 0
    my'' + ky' +mg = 0

    b) find the solution of the system of differential equations of part (a).


    2. Relevant equations



    3. The attempt at a solution

    there were no similar examples in the book so i dont know how to do part a... and for part b i tried:

    (mD^2 + kD)x = 0
    (md^2 + kD)y + mg = 0

    im not sure what to do from here or even if that was the right step to begin with
     
  2. jcsd
  3. Apr 13, 2009 #2
    The actual set of equations come from drawing a free body diagram of the projectile, and summing forces on the x and y axis (one equation for each axis).

    As far as solving the equations...i wrote a solution to a similar problem some time ago here on the forum. In it i used

    Force of air resistance = α * v,
    and for some reason i defined another constant B such that:
    B = α / mass

    And I assumed it started from a point (0,0) with velocity Vo. Aside from that it should get you on the right track..if not give you the answer :) (from https://www.physicsforums.com/showthread.php?t=241101 ). I also divided through by mass since it is a constant, then the differential equation is in terms of position/velocity/acceleration, something easy to interpret.

     
  4. Apr 14, 2009 #3
    thanks for the help

    this one was actually easier than it looked. all i had to do was find summation of forces for x and y direction and those were the 2 equations. then solving it was just a matter of doing both equations separately. i was trying to do it as a system of equations, but i dont think it can be done that way since there is no way to even check it as a system of equations
     
  5. Apr 14, 2009 #4
    Thats actually what I did. The two differential equations for the x and y axis are both independent of each other, and i solved them spearatley ;)
     
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