Coaster Dynamics: Tell Me More



Coaster Dynamics
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Roller Coaster Simulation Calculations
The motion of a roller coaster car is calculated by first developing a mathematical model of a roller coaster car. The mathematical model must take into account all the important physical laws and other factors relating to the system. To be accurate, the mathematical model must be detailed enough to include all the critical principles, and yet, simple enough to allow useful calculations.

In
Coaster Dynamics, the roller coaster car is modeled using equations of motion expressed in the form of "state space" equations. This is a special form of first-order differential equations (that is, it requires the use of calculus). The state space equations are solved on your computer using special mathematical "integration" algorithms.

For example, some simple state space equations might look something like this:

        dpy/dt = (mg)( cos²(ß) - 1 )
        ds/dt = (vy)/(sin(ß)).

Equations like these are derived for all the necessary physical parameters needed to model the car motion. When solved on your computer, these equations are converted in computer code, and calculated like this:

        dpdt = (mass*g)*( cos(beta)*cos(beta) - 1.0 );
        dsdt = vy*sin(beta);

Because of the complex physics involved, there are thousands of equations like these (except they are much more complicated!) that must be solved every second. For those familiar with numerical integration, the system equations are solved using a time step of 0.0001 seconds -- meaning that all the thousands of equations must be solved 10,000 times for each second of the simulation.

To learn more about these topics, see the Coaster Dynamics Physics Primer... particularly, Chapters 1, 4, and 8.