Since the Russian ice slides are the simplest form of roller coasters, we will start with them. First, the shape of the ice slides seemed to be a basic linear slope. They also tended to be a 50° drop, but they varied in height from 60 to 80 meters (Andrews, 2016). This means we can make a rough estimation of how long the slides were based on trigonometry.
First, since the angle we were given (50°) is complementary to angle θ, we know θ + 50° = 90° ⟹ θ = 40°. Then, we can use the trigonometric equation cosθ = adjacent/hypotenuse to find how long the slides would be: cos(40°) = (60m to 80m)/d. Solve for d and we have: d = 60/cos(40°) to 80/cos(40°) ≈ 78.32 to 104.43 meters. The slides would have been around 78 to 104 meters long.
In my introduction video, at the home page of this project, I explain how to find the speed that the rider is going at the bottom of the Russian Ice Slide. Here is an applet that will allow you to experiment with the size of the ice slide and allows you to see how fast the rider would be going at any point on the slide.
Moving on to the typical roller coasters we know today, many of the hills included in roller coasters can be interpreted as polynomials. Suppose we wanted to graph a roller coaster of our own by using a polynomial. Well, if we set a specific height to be our 0 point for y, then we could graph a roller coaster using the points where we want the car to pass that specific height as the zeros of the polynomial. For example, if we wanted to set the 0 point for y to be 30 meters high, we wanted to ride to start at 90 meters high, and we wanted two hills after the starting hill with increasingly smaller heights, we might create our polynomial with roots at 2 meters out, 10 meters out, 20 meters out, and the top of the last hill at 30 meters out. This would mean the polynomial would look like: y = a(x - 2)(x - 10)(x - 20)(x - 30)(x - 30). Keep in mind that the y-intercept we are looking for would be (0,60) due to the top being 90 meters and the “0 point for y” being 30 meters. Plugging that point into the function gives us a = -1/6000. Thus, our roller coaster follows the function y = (-1/6000)(x - 2)(x - 10)(x - 20)(x - 30)(x - 30).
In order to help you visualize this roller coaster, and create roller coasters of your own, here is an applet. Change up the function and watch how the acceleration due to gravity changes.