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Determining an Equation of Motion

  1. Sep 24, 2007 #1
    Problem Outline: I'm trying to determine how to keep the distance between 2 cars on a (3D) roller coaster ride. Currently the front car moves away from the back car.

    My current implementation uses a second order differential equation to model the acceleration of the cars at time t. The equation is as follows:

    y'' = ((n^3 * H^2 * y' ^ 2 * sin(n * theta) - n*g*H) * cos(n * theta)) / (R^2 + n^2 * H^2 *cos^2(n * theta))

    n = no. of hills,
    R = radius of the track,
    H = height of the hills (in metres)
    theta = angle of revolution about the track

    This ODE is numerically integrated using the Runge-Kutta method.

    Information much appreciated. Any further information please do not hesitate to ask.

  2. jcsd
  3. Sep 24, 2007 #2

    Chi Meson

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    I'm having trouble picturing what you want. What do y' and y" represent? Why would you cube the number of hills? and then multiply that by the square of the height of (which?) hill? What is theta measured from and how can that stay constant? Is the track circular, but with hills? More details...
  4. Sep 24, 2007 #3
    Hi Chi,

    Sorry for my explanations. I'll try to make it clearer. Basically what that second ODE does is it models the acceleration (y'') of a roller coaster car (theoretically it can represent any object) on a circular track that has radius (R), (n) hills each of height (H). Note the simulation runs in 3D. The angle (theta) represents the angle (in radians) that the car has with the x-axis. The car moves around the track in a clockwise manner. The (y') component is an approximation of the velocity (at time t).

    What I what to achieve is to maintain the distance between two cars at each time step. Since currently as the simulation progress the roller cars as they move down a hill they move a part and as the to the bottom they move closer. The same happens as the cars move up a hill and reach the top.

    If you need any further information please do not hesitate to ask :).
  5. Sep 25, 2007 #4

    Chi Meson

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    If you mean to say that you want to keep the distance between the cars constant, the only way is to have the cars always at the same point on a hill as the other. The hills would have to be equally spaced, and the separation of the cars must be some multiple of the separation between two adjacent hills.

    If you need a function to describe their separation as a function of time, someone else would have quicker insight to that than I.
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