On A Roll |
Roller Coaster Designer
Applications of Mathematics 10 |
Lesson Idea by: David Ward, Rutland Senior Secondary School, Kelowna, B.C.
The intricacies of a roller coaster are almost impossible to fathom. Not only are coasters highly technical and complex, but safety is a major concern, not to mention the laws of physics that must be employed.
"Roller coaster designers use math everyday," says Ron Toomer of Arrow Dynamics Inc. of Clearfield, Utah. "We use everything from straightforward high-school math such as angles and trigonometry; to physics math -- such as the laws of motion -- and, high level engineering math."
Mathematics allows the designers to ensure the safety of the roller coaster 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 roller coaster car going 135 kilometres an hour or plunging the height of a 22-storey 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 roller coaster designers!)
You're a member of a roller coaster 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 (computer-aided 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 metres from the loading station at a height of 5 metres above the ground. The top of the climb is located 40 metres away from the start of the climb at a height of 25 metres 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.
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 draw the above slope.
- Ground level is bottom line of grid, and acts as the x-axis.
- Start to draw the slope at the left-hand vertical line of grid, which is the y-axis.
- Draw a scale outline of the rise and run legs of the slope for the roller coaster.
- Label the various points as: x1, x2, y1, y2.
(Hint: Start drawing at least two squares up from the base line of grid, thereby allowing for the 5 m height of loading platform above ground level.)
Use this formula to calculate the slope.
Slope equals rise divided by run
M = (y2-y1)/(x2-x1)
In this case:
M = (25-5)/ (40-0)
M = 20/ 40
M = 0.5
Is this answer the same as your Rise/Run calculation?
Now, determine the mid-pint of the slope. Draw a vertical line from the mid-point of "run" leg (Hint: halfway between x1 and x2) to a point directly above it on the slope. Manually measure distance from ground to mid-point of 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 up slope from "mid base-point")
M (slope) = 0.5x = 20 (halfway along base line)
b = 5 (to represent that the slope starts 5 m above ground, or "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 the value you calculated manually?
"Designers clearly need to be able to calculate slope when they're working on a roller coaster, " 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 roller coaster. Draw the slope on the grids, and then calculate the mid-points 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 2m off ground
d. Slope from position 5 to 4
x = 20 m, ht. = 15 m
Start of run is 7.5m off ground.
e. Slope from position 8 to 7 (complete calculations to three decimal places)
x = 15 m, ht. = 7.5 m
Start of run is 2.5 m off ground.
Solution:
a. Slope .52, mid-point 11.5 m above ground
b. Slope .5, mid-point 7.5 m
c. Slope .25, mid-point 3.75 m
d. Slope .375, mid-point 11.25 m
e. Slope .333, mid-point 5 m (4.998 m)
Roller coasters 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 roller coaster affect the feel and intensity of the ride to make it fun?
Predict the optimal slopes for upward and downward sections of a roller coaster. How would you sequence the climbs and drops of a roller coaster to optimize the enjoyment of your riders? Remember, a roller coaster must be safe, as well as fun.
Sketch your own roller coaster, with a minimum of five 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 roller coaster, 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 per cent slope? (Hint: For every approximate 10 feet forward travel you drop one 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 five feet over a linear distance of 12 feet. What use is a pitch of 6/12? 8/12? 12/12?)
- Civil engineers have to carefully follow standards involving slope. When designing a roadway, what use would they have for a 30 per cent slope?
| Course/Grade: Applications of Mathematics 10 |
Curriculum Organizer:
Patterns and Relations; Shape and Space. |
Curriculum Sub-organizer(s):
Relations and Functions;
2D drawings of 3D objects;
and, graphing "slope." |
Prerequisites:
Understanding Ratios and Scale Drawings
Graphing of data
Working with formulae
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Resources:
· Graph paper
· Pencil
· Ruler
· Calculators |
| Student Activity Sheet - ON A ROLL
Participants: 1__________ 2__________ 3__________ Block:___ Quarter:___ |
Scale: 1 square = 2.5m (1mm = .5m) |
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1. |
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Slope = Rise/Run
Calculation of ht. at midpoint
y = Mx + b
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A. |
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Slope =
Calculation of ht. at midpoint
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B. |
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Slope =
Calculation of ht. at midpoint
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C. |
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Slope =
Calculation of ht. at midpoint
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D. |
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Slope =
Calculation of ht. at midpoint
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E. |
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Slope =
Calculation of ht. at midpoint
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