# The Darth Vader

### May the force be with you.

## The Vasu Group Ltd.

"The Vasu Group Ltd." is a rides manufacturing company based in Toronto, Ontario. It is one of the largest roller coaster manufacturers in the world and a market leader in the amusement industry. They do everything from designing to engineering as well as the manufacturing of various roller coasters. However, their most recent and most successful roller coaster project is "The Darth Vader", situated at Canada's Wonderland.

## History

The Darth Vader was announced on 23 November, 2013. The estimated cost to construct the roller coaster would be approximately $28 million. Located in Canada’s Wonderland, it is the 17th roller coaster built at the park. The Vasu Group manufactured this roller coaster through intensive designing, engineering and construction. In total, it took 18 months for the roller coaster to be made and opened to public. The coaster was officially in operation by May 5th, 2015. The name of the roller coaster was chosen to be "The Darth Vader" through the inspiration of the famed sci-fi series, Star Wars. There are a number of other roller coasters in the park such as the Leviathan and the Behemoth which are based on monstrous creatures. However, the creative team behind this project decided to do something unique by naming it as a character of the renowned Star Wars series. Besides the fact that it’s a simple yet memorable name, it is also an exceptional marketing strategy as the roller-coaster will create a lot of buzz among the large fan base of Star Wars.

## Specifications

Max Height: 300 ft

Min Height: 10 ft

Maximum Speed: 156 km/h

Model: Hyper-coaster

Duration: 1:40 min

Capacity: 40 people

## Table of Contents

2. Summary of Plan

3. General Descriptions of Equations

4. Time Calculations

5. Average/Instantaneous Rate of Change Calculations

6. Gallery of The Darth Vader

## Rough Draft

## Construction Plan

The creation of The Darth Vader involved several steps from designing and brainstorming to various mathematical calculations. Initially, we started by sketching a rough diagram of our roller coaster on paper. Our objective was to incorporate several different type of functions into our final design. We labelled different areas where we could place each type of function according to its appearance. Furthermore, during the design process we also had to consider approximately how much time each segment of the ride would last as per the restriction that the ride could only last for 100 seconds. Subsequently, we began constructing our roller coaster on Desmos. In order to construct our roller coaster, we had to incorporate horizontal/vertical translations as well as compression and stretches to align each function according to the design of the roller coaster. To obtain the fitting horizontal translations, we utilized the x-restriction value of the previous function. Moreover, in order to find the vertical translations we used the y-value of the last point on the previous function, and the y-value of the first point on the new function. We found the difference between these two values to find the exact c-value. Furthermore, in order to find the exact a-value of the next function, we substituted the last point of the previous function into the equation we created using the c and d values. We also utiized sliders to help make the functions fit our original design. Overall, throughout the design process, the main obstacles we faced were making all the functions perfectly align in sequence. Likewise, another difficulty we faced was making the roller coaster flow smoothly without sharp edges. In addition, we had to make the roller coaster appear to be realistic which allowed it to function in real life, thus making curves using quadratics was quite frustrating and time consuming. We made our transformations as accurate as possible by adding more decimals which was done through zooming in extensively into the graph on Desmos. Overall, we made the effort to make our roller coaster realistic, accurate and unique as possible.

## GENERAL DESCRIPTIONS OF EQUATIONS

The engineering and design of the roller coaster consists of multiple different kinds of functions. There are 8 main types of functions used in the design of this roller coaster.

Here is a breakdown of each different kind of function utilized:

*Let h represent the height. *

*Let x represent time.*

**Linear: an equation between two variables that gives a straight line when plotted on a graph. A linear is found in the form y= mx + b.**

**Quadratic: A quadratic function is a degree two polynomial function. A quadratic can be in many forms. One of the common forms is vertex form, which is expressed as y = a [k(x - d)]**²** + c.**

**Polynomial: A polynomial function is a function such as a quadratic, a cubic, a quartic, and so on, consisting of only positive integer powers of x.**

**Rational: Rational functions are a ratio of two polynomials. A rational function is found in the form y = [a / k(x – d)] + c.**

**Sinusoidal: A sinusoidal function includes the three trigonometric functions which are sine, cosine, and tangent. We only used the sine and cosine functions. A sine function is found in the form y = a sin [k(x – d)] + c. A cosine function is found in the form y = a cos [k(x – d)] + c.**

**Exponential: An exponential function is a constant raised to the power of the argument, where a and b are greater than 0, and b is not equal to 1. An exponential function is found in the form y = ab [k(x-d)] + c.**

**Logarithmic: A log function is the inverse of an exponential function. A log function is found in the form y = a log [k(x – d)] + c.**

**Square Root: A square root function looks like a half parabola. The base graph begins at the origin (0,0) and then moves upwards. A square root function is found in the form y=a√[k(x-d)] +c.**

## Time Calculations

## AVERAGE RATE OF CHANGE CALCULATIONS

## INSTANTANEOUS RATE OF CHANGE CALCULATIONS

## Photo Gallery

## may the force be with you.

## Group Members

- Ishu Singh
- Akashdeep Ghara
- Arvind Gogna
- Armaan Sidhu