Twists, Turns, and Thrills: The Math of Roller Coasters

Jun 22, 2022 | Novi

This summer, are you looking for a way to engage kids while applying math? How about amusement parks, specifically roller coasters?

Not everyone can be a NASCAR driver zooming around a lap at over 200 mph or an astronaut getting shot out into space at G-force but almost anyone can ride a rollercoaster and feel what it is like to push ourselves to the limit of speed.

Have you ever wondered what goes into building the perfect roller coaster experience? I’ll give you a hint, it’s a lot of math. Basic math subjects such as calculus help determine the height needed to allow the car to get up the next hill, the maximum speed, and the angles of ascent and descent. These calculations also help make sure that the roller coaster is safe.

The design of a roller coaster demands both creativity and thorough calculations. These calculations are strongly related to differentiation. For a thrilling ride, high speed and acceleration are needed. But for safety reasons, G- forces that people experience must be limited. Let's take a closer look at the math of extreme rides!


Firstly designers sketch a roller coaster showing the placement of hills, twists, and turns using math. Then they make a more technical drawing in a computer-aided design (CAD) program. This is where math becomes important. They have to calculate the slopes of 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 thrilling roller coaster designs incorporate 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 called a clothoid loop because it is shaped like a teardrop. 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. Clothoid loops in computer programs are modeled using advanced math functions.


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 energy 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.

So next time you visit an amusement park this summer, take a look around and appreciate the fact that every ride you see, depends on detailed mathematics. Pretty Cool.