On a Roll 
RollerCoaster
Designer

Lesson Idea by:
David Ward, Rutland Senior Secondary School, Kelowna, B.C. 

The intricacies of a rollercoaster are almost
impossible to fathom. Not only are coasters highly technical and
complex, but safety is a major concern. Don't forget the laws of
physics that must be employed.
"Rollercoaster designers use math everyday," says
Ron Toomer. He should know  he is a rollercoaster designer. "We use
everything from straightforward highschool math, such as angles and
trigonometry, to physics math, such as the laws of motion, and
highlevel engineering math."
Mathematics allows the designers to ensure the
safety of the rollercoaster while making the ride as enjoyable as
possible. Designers have to be precise when it comes to their
calculations. You probably wouldn't like hearing that someone had made
a mistake if you found yourself on a rollercoaster car going 135
kilometers an hour or plunging the height of a 22storey building.


In groups of 2 to 3, brainstorm and sketch as many
different examples of slopes as you can. For example, draw a roofline,
the trajectory of a thrown javelin, or a basketball free throw. For
each drawing, create labels for each component of the slope. Is there a
way to precisely calculate the slope you have created? If so, how would
you go about making these calculations?
Post each of the drawings around the classroom.
Use examples from each to explain technical terms such as "gradient" or
"incline." Also discuss the meanings of other associated terms,
including slope, slant, rise, run, ramp, pitch and plane.
Discuss situations in which slope and height are
involved. For example, why would a steeply pitched roof be a safety
hazard? Who else in the world of work has to know about slope? (Hint:
Think about people who design buildings and roads as well as
rollercoaster designers!)


You're a member of a rollercoaster design team.
The team has just completed the conceptual drawings for a new ride. It
looks really good and the client is pleased. The next step is to
transfer the design from freehand drawings into a CAD (computeraided
design) program. This will allow the machinists and technicians to
begin building parts.
In order to give the computer the necessary
information, you have to input some of your basic calculations. This
includes the slope of the first climb of the coaster. From your
drawings, you've determined that the climb starts 5 meters from the
loading station at a height of 5 meters above the ground. The top of
the climb is located 40 meters away from the start of the climb at a
height of 25 meters off the ground. The technicians will build the
climb portion of the track in halves, so you also need to determine the
height of the track halfway up the climb. (Note that your system of
measurement is in metric, for the sake of your international clients
who do not use the imperial system.)
What are the relevant points to use in
drawing the main slope?
What is the measure of slope?
What is the height of the track (from the ground) halfway up the slope?
Use square grid paper or the attached Student
Activity Sheet to draw the above slope.
 Ground level is the bottom line of grid. It
acts as the x axis.
 Start to draw the slope at the lefthand
vertical line of grid, which is the y axis.
 Draw a scale outline of the rise and run legs
of the slope for the rollercoaster.
 Label the various points as: x1, x2, y1, y2.
(Hint: Start drawing at least 2 squares up from
the base line of grid, thereby allowing for the 5meter height of the
loading platform above ground level.)
Use this formula to calculate the slope.
Slope equals rise divided by run
M = (y2y1)/(x2x1)
In this case:
M = (255)/(400)
M = 20/40
M = 0.5
Is this answer the same as your Rise/Run
calculation?
Now, determine the midpoint of the slope. Draw a
vertical line from the midpoint of "run" leg (Hint: halfway between x1
and x2) to a point directly above it on the slope. Manually measure the
distance from the ground to the midpoint of the slope. Record your
answer. Measuring manually is a lot of work and prone to error. A
faster, more accurate way of calculating the height of the track
halfway up the climb is done mathematically:
Use the equation for a straight line: y = Mx + b
Use these coordinates:
y = ? (height of track halfway upslope from
"midbase point")
M (slope) = 0.5x = 20 (halfway along base line)
b = 5 (to represent that the slope starts 5 m above ground or the "y
intercept")
Now, complete the calculation:
y = Mx + b
y = 0.5(20) + 5
y = 10 + 5
y = 15 m
How does this number compare to the value you
calculated manually?
"Designers clearly need to be able to calculate
slope when they're working on a rollercoaster," says Toomer.
"Sometimes it's to input into the CAD program, and other times to get a
feel for the design of the coaster."


Now you're ready to calculate the rest of the
slopes on the rollercoaster. Draw the slope on the grids, and then
calculate the midpoints for each slope:
a. Slope from position 2 to 3
x = 25 m
ht. = 18 m
Start of run is 5 m off ground.
b. Slope from position 6 to 7
x = 10 m
ht. = 10 m
Start of run is 5 m off ground.
c. Slope from position 9 to 10
x = 10 m
ht. = 5 m
Start of run is 2 m off ground.
d. Slope from position 5 to 4
x = 20 m
ht. = 15 m
Start of run is 7.5 m off ground.
e. Slope from position 8 to 7 (complete
calculation to 3 decimal places)
x = 15 m
ht. = 7.5 m
Start of run is 2.5 m off ground.
Solution:
a. Slope 0.52, midpoint 11.5 m above ground
b. Slope 0.5, midpoint 7.5 m
c. Slope 0.25, midpoint 3.75 m
d. Slope 0.375, midpoint 11.25 m
e. Slope 0.333, midpoint 5 m (or 4.998 m)


Rollercoasters are fun. That's why people build
them and that's why people ride on them. How do the upward and downward
slopes of a rollercoaster affect the feel and intensity of the ride to
make it fun?
Predict the optimal slopes for upward and downward
sections of a rollercoaster. How would you sequence the climbs and
drops of a rollercoaster to optimize the enjoyment of your riders?
Remember, a rollercoaster must be safe, as well as fun.
Sketch your own rollercoaster, with a minimum of 5 different slopes.
Calculate each slope. Write a rationale for which part of the ride
would be the most fun. (Note: The downward slopes will create a
negative slope.)
With your classmates, discuss negative slope. The
term negative slope is used to explain a downward trend.
 In the rollercoaster, what do you observe
about the slopes from positions 4 to 5, and 7 to 8?
 Is it correct to refer to a downhill grade in
a positive ("+") term?
 The next time you're driving through
mountainous terrain, look for road signs that indicate the grade of the
road ahead. What does it mean when the sign indicates a 10 percent
slope? (Hint: For every approximate 10 feet forward of travel you drop
1 foot.)
Here are some more topics to discuss with your
classmates:
 Why do different kinds of roofs have different
slopes? (Hint: For instance, the norm for a cedar shake roof is
expressed as 5/12. This means the roof rises 5 feet over a linear
distance of 12 feet. What use is a pitch of 6/12, 8/12 or 12/12?)
 Civil engineers have to carefully follow
standards involving slope. When designing a roadway, what use would
they have for a 30 percent slope?
Curriculum
Organizer:

Patterns, relations, shape and space 
Curriculum
Suborganizer(s):

Relations and functions;
2D drawings of 3D objects;
and graphing "slope" 
Prerequisites:

Understanding ratios and scale drawings
 Graphing of data
 Working with formulas

Resources:
 Graph
paper
 Pencil
 Ruler
 Calculators 
Student Activity Sheet
 ON A ROLL
Participants: 1__________
2__________ 3__________
Block:___ Quarter:___

Scale: 1 square = 2.5 m (1 mm = 0.5
m) 

1.


Slope = Rise/Run
Calculation of ht. at midpoint
y = Mx + b


A.


Slope =
Calculation of ht. at midpoint


B.


Slope =
Calculation of ht. at midpoint


C.


Slope =
Calculation of ht. at midpoint


D.


Slope =
Calculation of ht. at midpoint


E.


Slope =
Calculation of ht. at midpoint


Published in
Partnership by the Center for Applied Academics, Bridges
Transitions Inc., a Xap Corporation company and The
B.C. Ministry of Education, Skills and Training. Copyright
© 2002 Center for Applied Academics

