My team first sketches a roller coaster showing the placement of hills, twists, and turns. Then we have to make a more technical drawing in a computer aided design (CAD) program. This is where math becomes important. We have to calculate the slopes of our roller coaster hills in order to construct an accurate model; one that the construction crew can assemble correctly. Also, the slope will allow us to accurately determine the speeds that will be generated at various points along the track.
The most exciting roller coaster designs contain one or more loops. The loops must be built with extreme precision. A loop that is too circular will require very high speeds. This would result in a g-force that is too high for people to comfortably withstand. A perfect roller coaster loop is a teardrop shape called a clothoid loop. In a circle, the radius is constant but In a clothoid loop the radius changes and is shorter at the upper part of the loop than it is across the center. This means the roller coaster car can get through the loop at lower entry speeds. Advanced math functions are used to model clothoid loops in computer programs.
In order to complete a loop, a roller coaster car must travel at the correct speed. The speed that is needed is dependent on the size of the loop. Every roller coaster begins with a very high hill. The higher the hill, the greater the potential or stored energy of the roller coaster car. When the car reaches the bottom of the hill, the potential enery has been completely converted into kinetic energy which is the energy of motion. That's when the roller coaster car reaches its greatest speed.
To accurately model every component of roller coaster design, a branch of math called calculus is needed. Calculus is used to create and analyze curves, loops, and twists along the roller coaster track. It helps with slope calculations and finds the maximum and minimum points along the track.