# Design a Roller Coaster!

### By Karan, Purusothaman, Shamar & Moosa

## Our Roller Coaster!

## The Creation Process

Creating our coaster was relatively easy. Still, we did encounter many difficulties along the way. First, including 7 different functions in a certain Height vs. Time frame is easier said than done! With the help of the Advanced Functions course content, creating the functions themselves was the easy part. However, making them connect perfectly was difficult. It was increasingly frustrating connecting one function to the other, as we constantly shifted and stretched/compressed certain functions so that we could make them fit together. Placing restrictions on the time was also difficult as we had to take the maximum and minimum heights into consideration, as well as the time frame. Fortunately, we were able to perfectly connect each graph, leaving no holes (no matter how far you zoom into the points of intersection). Furthermore, achieving perfectly curvy parts in our coaster was one of our other main difficulties. Since we do not know calculus, we could not have perfect curves in our graph, but we still managed to make the final model much more realistic than our rough model. Overall, we designed a roller coaster that met the criteria of having a maximum height at 250 ft, a minimum height at 10 ft and a total length of exactly 100 seconds while using 9 functions.

## The Functions

## Function #1 - Exponential

The first function we decided to create is an exponential function. This function works perfectly for the beginning of a roller coaster as it shows that the roller coaster is increasing in steepness while the cart is accelerating. The restriction on time is {0<x<2} because the sudden change in steepness only lasts for a short period of time.

## Function #2 - Linear

The second function we used was a linear function. The function is perfect for this point of the roller coaster as it portrays the constant motion upwards that the cart of roller coaster is experiencing. The roller coaster is increasing in height at a constant rate while moving in a constant motion. The restriction on time at this part of the roller coaster is {2<x<20}. This point of this roller coaster lasts 18 seconds because for the roller coaster to run for a full 100 seconds, the cart needs to achieve an increased height so that there is more speed and velocity when the cart goes down the first 'drop'.

## Function #3 - Linear

The next function we used was another linear function. This function displays the roller coaster slowing down and stopping at the height of the first drop. Once the cart of the roller coaster reaches a height of 122.5 ft, it slows down and stops momentarily before it changes it's direction and goes in for the first drop. This process takes about 2 seconds which is why the restriction on time is {20<x<2}.

## Function #4 - Quadratic (Parabola)

The fourth function in our graph is a parabola. This represents the roller coaster going down the first drop and back up again. We choose this function because it shows how the roller coaster drops down and builds a high speed, and uses the momentum of that speed to go back up again within a short period of time. The restriction on time is {22<x<40} because although the roller coaster gains a lot of speed as it goes downhill, it eventually slows down a bit while going back up. The sharpness of the first drop was inspired by the initial drop of the 'Gravity Max' Roller Coaster, seen below.

## Function #5 - Sinusoidal

## Function #6 - Rational

We chose to use a rational function right after the sinusoidal function. The rational function portrays the roller coaster about to increase in speed while approaching another drop. The restriction on time is {60<x<75} because this part of the roller coaster illustrates the abrupt stop that it goes through which slows it down.

## Function #7 - Polynomial

The seventh function we used was a polynomial function. It shows another drastic drop in height and high speed the roller coaster experiences. After the drop, the roller coaster slows down when it reaches a height of 30m, then it slowly goes up and back down the small hill. The restriction on time for this function is {75<x<90} because after the increased amount of speed after the drop, the coaster slows down and then builds a small amount of height before going down the smaller hill.

## Function #8 - Logarithmic

The eighth function is a logarithmic function. It conveys the roller coaster moving slowly as the end of the ride is near. The restriction on time is {90<x<95} as this point is moving at a very slow rate.

## Function #9 - Linear

The function we chose to end off with is a linear function. This function shows the roller coaster decreasing in height at a constant rate and slowing down at a constant speed. This lasts for 5 seconds, so the restriction on time is {95<x<100}.

## Calculations

## Solve for the exact time(s) when your rollercoaster reaches a height of 250 ft:

## Solve for the exact time(s) when your rollercoaster reaches a height of 12 ft:

## Calculate average rate of change from 10 to 15 seconds

100/18 ft/second.