Mathematical
Exploration
“Exploring the
design of Roller
Coasters”
Introduction
About Roller Coasters:
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.
("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")
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 roasters. 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, foci, range and
so on.
("Photo TR: CW, La Ronde, & TGE 7/30-8/3")
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")
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.
("The Vertex of a Parabola")
("Using the Vertex Formula Quadratic Functions - Lesson 2")
Line of Symmetry:
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=
Foci:
-b
2a
("Axis of Symmetry of a Parabola")
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.
of a
("Mathwords: Focus
Parabola")
("Math TEKS Connections")
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)
Range:
The range of a parabola is all values of y from ( -¥, 𝑘 ], the y coordinates of the
vertices.
(Steege and Bailey)
Width:
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
Coster:(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.
4p(y - k) = (x - h)2
Using the formula above we can assume that the width of the parabola p is equal
to 50m. We can also place the vertices as 400 and 600 which can be seen as the
maximum of the coaster. 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 y=0.005x²+6x-1400 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.