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. Don't forget the laws of physics that must be employed.

"Roller-coaster designers use math everyday," says Ron Toomer. He should know -- he is a roller-coaster designer. "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 kilometers 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 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 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 2 squares up from the base line of grid, thereby allowing for the 5-meter height of the 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:

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 "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 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 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 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)

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.

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 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?

	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