Maryam Al-Naemi
Mathematical
Exploration
“Exploring the
design of Roller
Coasters”
Maryam Al-Naemi
Introduction
Roller Coasters (Background Information):
Most people have come to an agreement that the Russians were the first to
invent or conduct the mere idea of a roller coaster. In the 1600s they initiated the
first roller coaster in the form of ice that were converted into sleds, they even
placed fur or straw on the icy seat to make it more comfortable for the passenger.
The principle of friction was used to slow down the ride by placing sand on the
end of the ride to avoid it from crashing. As the years passed more advanced
sleds were made using wood to increase the thrill or intensity of the ride.
("Iceslide")
Roller coaster designs advanced and altered with time but a new era to its
design started when the first theme park in America opened: Disneyland. Until
this theme park opened all the rides were made using wood and this limited how
the loops on the rides were made. In 1959 the first tubular steel coaster was
introduced by Disney, he named it the Matterhorn. All the advanced features of
any thrilling roller coaster seen today such as a corkscrew track or loops are
tracked back to this roller coaster.
Maryam Al-Naemi
("A Look Inside the Matterhorn")
In 1992 the first successful inverted coaster was introduced, passengers
now have their feet dangling above them or even below them as the
circumnavigate the track. Six flags opened a coaster called Scream Machine; the
design of the coaster would have seemed impossible a few years ago. It is 415
feet tall and can reach a speed of 100 miles per hour, till today research and
technology towards more advanced ride designs are being made whilst working
with mathematical designs (properties of a parabola.) ("Amusement Park Physics -Roller Coaster")
("Richard Bannister")
Maryam Al-Naemi
Aim of the Investigation:
For all of these roller coasters there are many different types of designs. I
chose to investigate the math behind the second degree parabolic designs found
in most roller coasters. There are many reasons as to why this specific design
caught my interest. For one there is a balance of physics and math, as to how
the height might affect the velocity of the ride due to gravity and so forth. Also a
roller coaster has aesthetic features to it that make it more appealing to the
audience, and it intrigued me that a machines looks are sometimes as important
to its build. To conclude, skills that coincide with those of an engineer and an
architect would be needed to design this, both jobs that interest me. This is why I
chose this piece of investigation.
Here is the question:
To design a parabolic coaster you need to identify the properties from a
mathematical point of view; based on the required height, area and width. These
all come in place when the vertical parabola is designed with respect to the
mathematical aspects, which are, the: vertices, line of symmetry and foci. All of
these are placed aside to aspects of gravity and speed and put into
consideration. Everything is to contribute to the speed or thrill of the game in
attempts to make it more fun.
("Photo TR: CW, La Ronde, & TGE 7/30-8/3")
Maryam Al-Naemi
Investigation
The general equation of a vertical parabolic
curve:
The general shape of a parabola can be deduced mathematically
by the equation:
y = ax 2 + bx + c
𝑎, 𝑏 and 𝑐 are coefficients where a not equal to 0. Geometrically a
parabola is the set of discreet points in a plane and a given line.
There are two kinds of parabolas vertical or horizontal, I will
investigate mainly the vertical parabola. There are many reasons to
this one being that I make use of gravity and how it increases the
speed of the unit for the coaster. The different aspects of the
parabola can be seen in the image below:
("Parabola")
Maryam Al-Naemi
Vertices:
The vertex of a parabola is where it crosses its axis
(minimum/maximum point.) In most roller coasters the vertex is a
maximum point. This means the coefficient of x 2 which is 𝑎 is
negative. To acquire the vertices of a parabola (𝑥, 𝑦)
x=
-b
2a
To get 𝑦 substitute by the value of 𝑥 found using this equation in
the original equation of the parabola to get the 𝑦 coordinates. When
the vertex is obtained it can be written in the “vertex form”:
y = -a(x - h)2 + k
Where (ℎ, 𝑘) is the coordinates of the vertices. (Peak of the coaster)
("The Vertex of a Parabola")
("Using the Vertex Formula Quadratic Functions - Lesson 2")
Line of Symmetry:
Maryam Al-Naemi
The line of symmetry (axis of symmetry) is the line that runs
down its center and divides it into two equal halves. It can be
obtained mathematical using:
x=
-b
2a
("Axis of Symmetry of a Parabola")
The use of line of symmetry in building a roller coaster is very crucial.
For one it helps plan the width of the parabola in correspondence to
the ride as a whole. Also it avoids make the parabola in the ride too
high in order to avoid, flying objects, power cord poles or such. Also
for when there are pieces of metal above parabolic part of the ride it
helps avoid it being so high that the people playing may hit there
heads.
Foci:
The focus of a parabola is a fixed point within its interior. It must be
on the axis of symmetry line. Located with a fixed distance from its
vertex called P.
("Mathwords: Focus of a Parabola")
("Math TEKS Connections")
Maryam Al-Naemi
There are four different forms of expressing the focus of a parabola the standard
form being:
Focus of a Parabola
Vertical Parabola:
4p(y - k) = (x - h)2
(ℎ, 𝑘) are the (𝑥, 𝑦) coordinates of the vertex. The focal point is measured in 𝑝
units from the vertex on the axis of symmetry. (ℎ, 𝑘) are the coordinates of the
vertex thus 𝑝 would hold the coordinates (ℎ, 𝑘 − 𝑝)
The foci is crucial to build of the roller coaster in order to indicate the
interior width of the more parabola at the peak before the cart goes
from going up to going down. The bigger the foci the steeper the
parabola will be and the faster the cart would go.
A width of a parabola is the length of a horizontal line that passes through the
focus and touches the parabola at each end. This can be found using:
4P
(sites.csn.edu/)
Designing a Parabolic Roller
Coaster:(Autograph)
For most roller coaster a parabolic lift should be at a maximum height of
300-400 feet in order to be taken as a thrill ride. Each roller coaster has its own
mathematical aspects and sizes. For the benefit of research we will design a
parabolic roller coaster and examine the math behind it. An example of a
parabolic coaster of height exceeding 300 is Millennium Force in Cedar Point in
Ohio.
The coaster has to have a certain height in order for gravity to have a stronger
force and for safety reasons, it can’t go too high because the metals may not be
able to hold the cart or it might rot with weather changes. Also the cart size has
to be put in perspective for its weight and size has a place on how much the
coaster can hold.
4p(y - k) = (x - h)2
Using the formula above we can assume that the width of the parabola p is equal
to 50m. According to the available area used for the coaster. We can also place
Maryam Al-Naemi
the vertices as 400 and 600 which can be seen as the maximum of the coaster.
This is according to the survey data to the area. Meaning the axis of symmetry is
600. This means that (ℎ, 𝑘) are equal to (600,400).
WA=e can deduce from the graph below showing the equation
-4(50)(y−400)=−(x−600)² the x-intercepts as shown are 317.2 and
882.8.Now we examine the math behind this roller coaster, first lets determine
the height which can be taken from the vertices.
600
y
400
200
Point: (882.8, 0)
Point: (317.2, 0)
−200
200
x
400
600
−200
−400
−600
−800
Foci:
In our case the equation would be:
800
1000
1200
1400
Maryam Al-Naemi
Width:
= 4p
= 4(50)
= 200
Slope:
The slope can be determined in many ways although in order to gain
the maximum and minimum slope of the entire parabola we need to
find the first derivative. By knowing the parabolas maxima and mina
points of slope we can determine when the cart on the roaster
coaster will be at its steepest points. This information can help the
engineer working to make the coaster to study the speed of the
coaster and gravity’s effect on it.
-4(50)(y-400)=-(x-600)2
-200(y - 400) = -x 2 +1200x - 360000
1 2
y - 400 =
x - 6x +1800
200
1 2
y=
x - 6x + 2200
200
dy
1
=
x-6
dx 100
Now that we have the first derivate we can make it equal to 0 in order
to find the vertices which are the max and min points. Being a roller
coaster there are no minimum points, also the parabola is concave
down. Also finding the derivate could help find any inflexions found in
the parabola in order to make the build more accurate. Also by doing
so we can see if the coaster has more of a sin or cos build or a
parabola.
Maryam Al-Naemi
dy
1
=
x-6= 0
dx 100
x = 600
Examine different Types of Roller
Coasters:
Designs a Coaster might have other than
Parabola.
Circle Ellipse:
Circles are used as one of the elements of a coaster, there are full
circles and irregular circles depending on the coaster. The equations
used to deduce the circle are by the circumference (C) , which is the
distance around the circle. Based on the available area of the floor
helps me decide the area of the circle itself of the coaster for the
circle, also its circumference helps determine how many carts would
be in the coaster. This also helps me have to determine the power or
requirements of the engine and want power it has to exert to make
the cart go around the circle.
("Math Applications Ch 3 Curves")
Maryam Al-Naemi
As an example some coasters have a radius of about 13m which
makes the circumferences equal to 81.7m so this gives us the
required length for the circle, things such as carts limitation. The area
would be 530.9m2 so according to this area we can decide on the
logistics of the coaster.
("Roller Coasters and Amusement Park Physics")
The equation of a circle is equal to
where r is the radius.
("How Roller Coasters Work")
x 2 + y2 = r 2
Sin Curve
Some roller coasters use a sin curve as a part of there coaster. The
equation used is:
𝒚 = 𝒂 𝒔𝒊𝒏 (𝒃𝒙 + 𝒄)
𝒂 is the amplitude of the sine curve
𝒃 is the period of the sine curve
𝒄 is the phase shift of the sine curve
For example a sin curve for a coaster could have an amplitude of
25m and the period as 2𝜋 and the phase shift as 0. So we come to
Maryam Al-Naemi
an equation of
𝒚 = 𝒂 𝒔𝒊𝒏𝒙
𝒚 = 𝟐𝟓𝒔𝒊𝒏𝒙
("Assignment 1: Exploring Sine Curves")
Maryam Al-Naemi
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Maryam Al-Naemi
"Photo TR: CW, La Ronde, & TGE 7/30-8/3." Themeparkreview. N.p., n.d. Web. 6 July
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