Roller Coasters Research Paper

Roller Coaster Math
Motivation:
I’m intrigued by roller coasters and I’ve a personal interest in theme park engineering, a fusion of both technology and imagination. I’ve studied and considered how engineers innovate new and exciting theme park rides and attractions, design and model them to be appealing and safe, determine how they will be built, and manage their construction.
Designing roller coasters, which are exciting and safe, is both an art and science. My goal is to explore the mathematics behind the design of a drop (for thrill) and loop (for weightlessness) roller coaster where rider can go down a big drop and speed through a loop upside down that is both scary and safe till the end of ride. We can think about the track consisting

The greater the centripetal acceleration, the greater the G-Forces felt by the passenger. Earth gravitational force is 1G, also described as their normal weight. Weightlessness can be described as 0G. When one goes downhill on a roller coaster they experience a G force between 1 and 0.
The Drop:
My first goal is to determine the G-forces experienced at the bottom of the first drop of my new roller coaster.
I have taken the height of my coaster to be 91.0m which is comparable to some of the fast and thrilling coasters of today and I have modeled the drop section using the following function which starts at a height of 91 m at x = 0 and drops to ground level at x = 85π/2: y(x) = 91 e^((-x)/500) 〖cos⁡〖x/85〗〗^2 where 0≤x ≤85π/2
We represent the track height in m as y(x)and look what happens in the first 135 m of horizontal distance.
We know from the conservation of total energy that the velocity is: v(x)=√(2(91-y(x)) ,since 1/2 mv^2=mg (91-y(x))
We know that the centripetal acceleration is be a_c= v^2/r, where r is the “radius of curvature” at the point where we want to measure centripetal acceleration.
Radius of Curvature and Arc

The reason for this is that it is desirable to avoid discontinuities in the force exerted on the car. Since a component of that force comes from centripetal acceleration, which is directly proportional to the curvature, the curvature must remain continuous. A discontinuity would not only be uncomfortable for the rider. We want a curve that is fun to travel along, which means sharp curves—but not too sharp, unless we want our amusement park guests to get sick or pass out.
The distance a roller coaster car has traveled along the track will be denoted as S, while the distance a car has traveled into an element will be denoted s. The total length of an element is denoted by s_i where the subscript i denotes the number of the element in the ordered set.
The Basics: Loops with constant g-force
Loops with constant g-force
Another possibility to avoid the sudden onset of large g-forces, could be to design a loop with constant g-force, either throughout the loop, or through part of it. Let the condition be applied below a point where the track forms an angle 0 with the horizontal, has a local radius of curvature r0, and the centripetal acceleration is, again, given by

a_(c,0)=(2gh_0)/r_0

The g-force factor is given by the force from the train on the rider divided by the weight of the rider, which can be expressed as a