Transformation to a Stationary Frame:

As noted the accelerometer data describes the roller coasters motion in terms of the acceleration due to the track and relative to an observer riding in the car.  If we could translate this data into net accelerations relative to a stationary observer outside the system, we could phrase a discussion in terms similar to those used during in-class examinations of motion.  This would also allow us to examine velocity, position, and energy.   This proved to be a daunting task.

The primary obstacle was the lack of independent data.  A coaster moving in 3 dimensions requires a minimum of five values to completely describe its motion relative to a stationary observer: the accelerations due to applied forces along the 3 axis and two angles describing its orientation.  The data provides information about accelerations but none about orientation.

The solution to this problem is to combine the measurements with the constraints of the system. The motion of the coaster can be defined piece wise depending on weather their is a radial acceleration.  The mathematics of the 3-D case proved to be far beyond the scope of a introductory physics  audience. Fortunately Montazooma's Revenge is essentially restricted to two dimensions.  If the movement of the coaster is confined to the x,z plane, all radial acceleration is along the z-axis. Further, any radial acceleration implies a changing normal force due to the changing component of gravity in the z-direction.  Similarly, a constant normal force implies no radial acceleration.

Therefore:

Finding the angle when there is no radial acceleration is relatively straight forward.  Finding it when the track is curved is not.  If we substitute an expression for radial acceleration in terms of velocity and radius into the above equation we get:
 
 



 


Substituting for known quantities and angle as a function of time results in:
 



 


While it may be possible  approximate Ax(t) and AN(t) and apply differential calculus to solve for angle as function of time, the techniques needed are far more advanced then plausible for an intro level class. A more feasible method is to step through the data assuming constant accelerations over small intervals of time. Given that we know the initial conditions and we can  approximate the angle after a short time, using only linear and rotational kinematics.

The next angle would be given by:
 


Therefore we need the  angular acceleration and velocity at a given point.

The velocity tangent to the track is given by:
where D vg is the change in velocity due gravity and D vAx is the change in velocity due to any other force tangent to the track such as friction.  The change in velocity due to gravity can be approximated by averaging the components of gravity at the current point and and the previous point and multiplying by the time interval.
Similarly the change in velocity due to all other forces and be found by calculating the area under the curve of  Ax vs. Time.  Because there is no data between points, the closest approximation of the area is simple the average reading multiplied by the time interval.
Combining the expressions we get:

and

Using these expression we can solve for the velocity a each point and in turn the angle.

The next step is to combine the angle with the measured accelerations to find the components of the net acceleration relative to a stationary coordinate system.
With both the tangential velocity and accelerations relative to a stationary observer, a more comprehensive study of the physics behind the coaster's motion is possible.