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Euler's Method for motion of golf ball with lift

  1. Apr 19, 2009 #1
    1. The problem statement, all variables and given/known data

    We were given two equations of motion for a golf ball with lift. We must figure out the maximum range using an initial velocity of 255 ft/sec. I'll just retype what's on the sheet.

    "Lets incorporate lift. The free body diagram with all the forces shown is depicted in figure 3. Recall lift is a linear function of velocity, CL = k/V, k is a constant. The ration of k/m, where m is the mass of the golf ball, has been found experimentally to be 0.247 sec^-1. The equations of motion are now

    1. -cVx - kVsin(angle) = mdVx / dt = -0.25Vx - 0.247Vy = dVx / dt
    2. -cVy + kVcos(angle) - mg = mdVy / dt = -0.25Vy + 0.247Vx - g = dVy/dt

    These are now coupled equations where Vy depends on Vx and Vx depends on Vy. Sorry there is no simple analytic solution to these equations. So we will solve them numerically. HOwever, this is crucial, we do have an analytic solution for k=0. Thus we can check our numerical solution with the k=0 case and see if we recover the analytic solution. This is a standard technique in science and engineering, always check a numerical solution against an analytic solution. Then you can have confidence in your numerical solution. The simplest way to solve equations like the ones we have is the Euler method. Basically you break the solution into small steps, where the solution at the nth step depends on the solution at the nth-1 step. You should be able to implement on Euler solution on any PC. You can adjust the time step and see how the results depend on the choice of chance in t. Compute numerical solutions for angles starting at 4 degrees to 16 degrees, say every 2 degrees and determine the maximum range using an initial velocity of 255 ft/sec. Once you know how to determine the maximum range, find the difference in range between Atlanta and Laramie. You will have to track down the difference in air density between the two altitudes."

    2. Relevant equations

    1. -cVx - kVsin(angle) = mdVx / dt = -0.25Vx - 0.247Vy = dVx / dt
    2. -cVy + kVcos(angle) - mg = mdVy / dt = -0.25Vy + 0.247Vx - g = dVy/dt

    3. The attempt at a solution

    All we have got is the Euler equation for this because he helped us with it.

    dx / dt = V
    xi+1 - xi / ti+1 - ti = Vi
    -0.25Vx - 0.247Vy = dVx/dt

    Vxi = t1 - t0 [f1(Vxi, Vyi) + Vxi)] *So I think this is the Euler's equation for this. I kind of understand how to do Eulers, but not on equations like this. If anybody can help please do. Thank you.
     
  2. jcsd
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