Algebra 2 Roller Coaster Design Project
Your engineering firm (your group!) has been asked to design a roller coaster for an amusement
park to be built adjacent to the Tuscola Outlet Mall. Several other firms are expected to submit
designs; the design that provides the most "thrill" and "eye appeal" while providing safety and
incorporating siting restraints will be selected. The selection committee has broken the design
process into several phases, each with a deadline and specific requirements. As you complete
each phase you will be critiqued so that you can incorporate necessary changes into the final
design.
Some of the constraints identified by the selection committee include:
•
The roller coaster must fit on a 35 acre plot of land with no "holes" (you must count any
land lying between sections of track, or lying within loops, as part of the 35 acre plot),
BUT you may choose any dimensions for the plot.
•
The ride should last about two minutes.
•
The ride may have only one lift motor and one braking section.
•
The track may be metal or wood, but the "train cars" must be conventional (that is, they
must ride atop two parallel rails, and the rails are 6 feet apart [2 m. if you wish to use
metric units]).
•
The roller coaster ride must be on a track that is a closed circuit; that is, the train travels
along the track in one direction only, and returns to the station along the circuit to change
passengers (some recent roller coasters travel forward along the track until it ends, then
roll backwards into the station--yours CAN'T DO THAT).
•
The ride must be SAFE.
Here are the phases of the design process:
Phase 1:
Form a "company" and create a (very basic!) design
Phase 2:
Analyze a coaster "hill"
Phase 3:
Analyze the coaster support structure
Phase 4:
Staying on track at the tops of hills
Phase 5:
Loops and curves: adding "Thrill"
Phase 6:
The "Roller Coaster Scream"
Phase 7:
The Final Design
Suggested Links:
Roller Coaster Photos and Descriptions
Roller Coaster Interactive Sites
•
Coney Island "Thunderbolt"
•
Physics simulator applet (Java)
•
... different view
•
•
More Coney Island
"Build a Coaster" (Discovery
Channel)
•
Banked curve on Cyclone (Coney Island)
•
Video Clips (Discovery Channel)
•
"Wonder Wheel" (Ferris-wheel/RC
combination?)
•
"Wild One" at Six Flags America, Largo,
MD
•
Lots of parks...
•
"Bobcoaster" review of Six Flags, St. Louis
•
"Bobcoaster" home page
•
Mine Diver--technical data from the
manufacturer of this ride in a park in South
Africa
•
Roller Coaster Data Base: a fairly thorough
listing of roller coasters around the world.
Contains statistics (height, length of track,
duration, etc.) and photos.
Phase 1: Start a new "company" and create a (very basic!) design
Some background information:
•
In basic designs, the only energy applied to the "train" (the set of cars where the riders
seek thrills) is applied as the train is pulled up the first hill by motors. The laws of
physics take over after than: potential energy at the hilltop is converted to kinetic energy
at the bottom. Some energy is lost due to friction as the train careens around the track,
and there is a braking system (more friction!) to slow the train to a stop when it reaches
the station.
•
For the gravity constant g, use 32 ft/sec2 and work with feet as your unit of distance (in
metric, it's 9.8 m/sec2).
•
The steepness of the down-side of hills determines how quickly the coaster picks up
speed: the steeper the hill, the greater the acceleration.
•
Tall heights, sudden drops, tight turns, high speeds, and "sensory effects" (such as
tunnels, water spray, sound effects, lighting) are what give roller coaster riders their
thrills… the more, the better!
•
One acre is 14,560 ft2.
Your Requirements:
1. Decide on a name for your company. You may use this name for your roller coaster, but
you don't have to. Make a company logo to affix to a folder where all your work will be
stored.
2. Examine some photos to get some ideas of what typical support structures for roller
coaster track look like, and some idea of how long the track should be (you're shooting
for about 2 minutes of fun). You'll also want to look for some special features (or make
up your own!) to make your coaster more exciting.
3. Decide on a basic layout for your roller coaster. You'll need to figure out the dimensions
of the land required for your roller coaster (35 acres, remember!) and work within that
area. Make a scale drawing (top view) of your layout. This drawing can be a fairly rough
sketch; you'll fill in some of the details later. Be sure to indicate the scale you're using
somewhere on the sketch. Keep in mind that the actual track will go up and down, so it
will be longer than the track that you draw on your layout; you may assume that it is 2030% longer.
4. Decide on a "profile" for your roller coaster. Determine the number and height of hills,
and make a drawing (as seen from one side) of a representative hill, showing the support
structure. On your side view, show the ascent angle and the drop angle clearly (both are
measured from the horizontal).
5. Will the train stay on the track you designed? Identify places on your scale drawings for
parts 3 & 4 where this question might be significant.
Phase 2: Analyze a coaster "hill"
Some background information:
•
The Mean Streak roller coaster was built in 1991 at Cedar Point Amusement Park in
Ohio. The total length of track is 5427 feet. The train climbs up 161 to the top of the first
hill, then drops 155 feet at a drop angle of 52° . The track has banked curves and
crisscrosses the support structure nine times. There are 12 hills and valleys on the coaster;
the ride lasts 2 minutes 45 seconds, and the top speed is 65 mph. Mean Streak's second
hill is 123 ft 11 in. high.
•
In 2000, Cedar Point opened Millenium Force, the world's tallest and fastest roller
coaster. Its first hill is 310 feet high, at a climbing angle of 45o, and the train climbs it at
a speed of 20 ft/sec (total time: 22 sec... is that correct?) then drops 300 feet at an angle of
80o, attaining a speed of 92 mph.
•
The Texas Giant at Six Flags Over Texas opened in 1990. Total track length is 4,920
feet. The first hill is 143 feet high, with the first drop at an angle of 53o for a total drop of
137 feet. The ride takes 2 minutes, and there are three 28-passenger trains running on the
track at any one time. Top speed for the coaster is reported to be 62 mph.
•
One of the main ideas of physics that makes roller coasters work is CONSERVATION
OF ENERGY. We can simplify the situation and assume that the roller coaster has either
potential energy or kinetic energy or some combination of the two. "Conservation" means
that once energy is placed into the system (in the case of a roller coaster, the motor that
pulls the train to the top of the first hill places energy into the system), the total energy in
the system remains constant unless work is done (we'll assume that the braking at the end
of the ride is the only work done). Thus, PE + KE = TE, or potential energy + kinetic
energy = total energy. Potential energy is given by PE = mgh, where m is the mass of the
train, g is the acceleration due to gravity (use 32 ft/sec2 and work with feet as your units),
h is the height above the station. Kinetic energy is given by KE = mv2/2, where v is the
velocity of the train in ft/sec. Thus, the "total energy" equation can be re-written, using
these formulae, as mgh + mv2/2 = TE = constant.
•
You may assume that the motor that pulls your train to the top of the first hill can pull at a
constant speed of 4 feet per second. The time it takes your train to get to the top of the
hill will depend on the length of track between the bottom and top of that hill; you can
use the "constant velocity" model rate*time=distance to calculate the time. Since your
ride is only about two minutes, you might not want more than about 30 seconds of
climbing…
•
For all remaining hills, up and down, the time it takes the train to traverse the slope may
| Δv |
be approximated by t =
, where t is the time in seconds, Δv = vbottom – vtop =
g sin θ
(velocity at the bottom) - (velocity at the top) [both can be obtained from the total energy
equation], and θ is the angle of ascent (climb) or drop angle.
•
Time spent on horizontal portions of the track comes from your old friend,
rate*time=distance.
Your Requirements:
1. Make a scaled sketch (full sheet of graph paper) of your first hill; you may wish to base it
on one of the roller coasters in the background information (you may just copy the first
hill from one of these rides, if you wish). Label La as the vertical (ascent) height of the
hill, θa as the ascent angle, Ld the vertical drop of the hill, and θd as the drop angle. Note
that the "down sides" of both Mean Streak's and Texas Giant's hills don't make it all the
way to the bottom. Label Lh as the horizontal distance (length) from the base of the hill
on the "up side" to the bottom of the first valley. Use trigonometry to calculate the value
of Lh.
2. Use trigonometry to calculate the length of track in the first hill (from the base of the hill
on the "up side" to the bottom of the first valley).
3. You don't know the mass of your train, so you can't write a numerical expression for the
total energy at the top of the hill. You do, however, know the height of the hill, and you
know g and your train's velocity at the top of the first hill (4 ft/sec). Use this information
to write an algebraic expression for the total energy at the top of the hill that involves
only the mass as a variable. You'll want to circle this expression, because from now on it
gives the TOTAL ENERGY in your roller coaster system.
4. Write an expression for the total energy of the train at the bottom of the first hill (be
careful as you consider the value to use for h!). This expression will have two unknowns:
m and v. But, due to conservation of energy, the total energy at the bottom equals the total
energy at the top! Use this fact to eliminate m and solve for v. Convert your answer for v
into miles per hour to compare with the information given for Mean Streak or Texas
Giant.
5. Does your sketch for item 1 have "corners" at the top and bottom of the hill? Do roller
coasters have such corners? How do you think you should modify the sketch to make for
a "smooth" ride? Discuss…
6. Calculate the total time your train will spend on the first hill (total time = time UP + time
DOWN; see background information).
Phase 3: Analyze the coaster support structure
Background Information
•
Triangles are often used in construction because they're rigid. As long as the beams used
in construction, and the joining mechanisms (bolts, welding, etc.) don't fail, triangles
won't collapse. Think about it: a three-legged stool can't wobble, but a four-legged one
can; if you make a triangle out of popsicle sticks attached with a thumbtack at each
corner, the triangle won't deform, but if you make a square out of popsicle sticks you can
deform it into a rhombus...
Requirements
1. Select a picture of a roller coaster whose support structure is clearly visible. Find two
sections of the support structure that contain triangles that are not right triangles.
2. Obtain an enlarged photocopy of each triangle, and highlight the triangles you’re using in
color. Label the first triangle’s angles A, B, and C, and label its sides a, b, and c
(according to the "usual" convention: side a is opposite angle A, etc.). Measure angle A
and sides b and c on your drawing (show the measurements on the drawing).
3. Use the law of cosines and/or the law of sines to "solve" your triangle (predict the length
of side a and the measures of angles B and C). Measure side a and angles B and C;
comment on any discrepancies.
4. Label your second triangle’s angles J, K, and L, with sides j, k, and l. Measure side j and
angles K and L.
5. Use the law of cosines and/or the law of sines to “solve” this triangle. Then measure to
confirm your work. Explain any differences [note to any lazy engineers: the "selection
committee" will carefully check your calculations and your measurements if you claim
that your predictions were correct in order to avoid trying to explain any differences].
6. Be sure to cite the source of the photo you use (not only the book or web site, but, if
possible, the names of the park and the ride).
Phase 4: Staying on track at the tops of hills
Background Information
•
For objects acted upon by gravity (such as the roller coaster train!), we can be more
precise with the velocity part of the "conservation of energy" equations. If you think
about an object moving on a ramp, its velocity at any time tells how fast the object is
moving along the ramp (or track!). If you think about the motion, it is partly up or down
(vertical) and partly forward (horizontal). A right triangle can be drawn to represent the
velocity: the hypotenuse is the actual velocity, the base is the horizontal part (or
component) of the velocity (which we'll call vh) and the altitude is the vertical component
(vv). Just as with a plain, vanilla right triangle, the Pythagorean Theorem applies, as does
right triangle trigonometry--we can write v2 = vh2 + vv2.
•
Since gravity acts vertically, only the vertical motion of the object is affected by gravity;
the horizontal motion is not. If the train were "launched" (like a cannonball, without the
track), then its vertical height above the launch point would be given by y = vv t - 0.5gt2,
and its horizontal distance from the launch point would be x = vht . Of course, the
vertical and horizontal components of velocity at the time of "launch" depend on the
velocity v at launch time.
Requirements
1. Consider the angle of ascent for your roller coaster's second hill as the "launch angle."
Sketch a right triangle for the velocity components, as suggested in the background
information. Label the angle at the base of the triangle (which corresponds to your angle
of ascent) as θ, and use trigonometry to write expressions for vh and vv in terms of θ and
v (which comes from the Total Energy equation!).
2. The graph of y (height) as a function of t, from the background information, is a parabola
that opens downward. Thus, its vertex is a maximum height. If your second hill isn't that
high, then the train will actually leave the track at the top... not a very safe maneuver.
Find the y-coordinate of the vertex of the parabola y = at2 + bt (in terms of a and b; refer
back to the Algebra 2 textbook's unit on quadratic functions if you have to!), then express
the vertex of the height equation for your roller coaster in terms of vv and g (in this case,
a = -0.5g, b = vv). Use your results from item 1 to express this minimum height for hill
#2 in terms of v, g, and θ.
3. The maximum height for hill #2 is the same as the height for hill #1, and this would result
in almost all of the energy in the system being potential energy, which was the case at the
top of hill #1. That would mean that the coaster would nearly stop (4 ft/sec) at the top of
the hill... not very exciting. At the top of hill #2, in fact, you'd like to have at least some
velocity. Design your second hill carefully now, taking into consideration the train's
velocity at the bottom of the valley right before beginning the climb, the speed you'd like
to maintain at the top of the hill (which, using the TE equation, determines the height of
the hill), and an ascent angle that will allow your coaster to stay on the track. Make a
scale drawing of your second hill (indicate ascent and drop angles, as well as height, on
your drawing, and attach all calculations). FOR EXTRA THRILLS (optional): choose
the ascent angle so that the height of the hill is the minimum height for the v you’re
using… riders will feel weightless at the top!
4. Calculate the time spent going up and down hill #2 (refer to Phase 2 for the equations for
timing).
5. (OPTIONAL) Solve the equation of horizontal position given in the background
information for t, then substitute your result into the expression for vertical position. If
you also express vh and vv in terms of θ and v, you will end up with an equation for the
parabolic path the train would take if it were launched. You "might" want to make sure
that your track stays under the train: the parabolic path should lie below your track.
6. (OPTIONAL) The idea of "launching" the train from the track is a serious consideration!
At the crest of every hill, the track's curvature (remember, no sharp corners!) must be
more gentle than the parabolic path the train would take if it were "launched" horizontally
(angle of 0o) with the velocity v given by the total energy equation for the top of the hill.
On your design for item 3, show the curve at the top of hill #2 and verify that the
parabolic 0o launch path lies below your track.
Phase 5: Loops and curves: adding "Thrills"
Background Information
•
Ever hear of "centrifugal force"? Well, there's really no such thing. When you feel
pushed to the side as a car makes a fast turn, what's really happening is inertia, another
law of physics. Your body wants to travel in a straight line, but the car is forcing it to
turn. Inertia is what forces you to lean into the curve when you're riding your bike
around a corner; if you didn't, your bike would turn (the friction of the wheels on the road
forces it to) but your body wouldn't, causing separation of body from bike. Well, if
you're wearing a helmet, you'll probably survive... but we certainly DON'T want any
riders thrown off the roller coaster, and we don't want the coaster to jump track or to wear
out the flanges on its wheels. Banking the turns will achieve
this! The banking angle α (measured from the horizontal) is
v2
given by the equation tan (α ) =
, where v is the velocity
Rg
of the train, R is the radius of the curve, and g is the
acceleration of gravity.
•
If you'd like your roller coaster to have a loop, you have to fight that parabolic launch
tendency you worked with in Phase 4. This time, since at the top of the loop the track is
above the train, the track must lie below the parabola. Fortunately, there's a fairly simple
way to achieve this. For a circular loop, as long as the radius of the loop is less than
vT2/g, where vT is the velocity of the train at the top of the loop, the train will stay with
the track.
Requirements
1. Return to the layout for your roller coaster (from Phase 1). Select one of the curves (if
you actually put corners on your track, you need to go back and correct that problem!).
Make a (larger) scale drawing of the curve. Assume that the curve portion is an arc of a
circle; use a compass to draw that part. Measure the radius of the curve to scale (feet).
2. Determine the angle through which the train turns on that curve in two different ways:
First, draw tangent lines at the start and finish points on the curve. Extend the lines until
they intersect; measure the angle of intersection (you decide whether to use acute or
obtuse angle…). Second, draw radii through the start and finish points, then measure the
central angle formed by the two radii. Are the two measurements the same?
3. Determine the elevation of the curve (you may just assume an elevation, consistent with
the curve’s location on your track layout), then use the Total Energy equation to
determine the velocity of the train on that curve.
4. Determine the appropriate banking angle for the curve. If you're going to make an error
on this calculation, it's better to make the angle too small, and rely on the wheel flanges
on the cars to keep the train on the track, than too large, causing the track to dump the
entire train into the center of the curve!
5. Find “thrill” features on four different roller coasters. Be sure to cite your sources, in
addition to naming the park(s) and the rides.
6. (OPTIONAL) Design a loop into your roller coaster. Make sure you use the Total
Energy equation to determine the train's velocity going into the loop! Make a scale
drawing of the loop.
7. (OPTIONAL) Most roller coaster loops are NOT circular, they're more "teardrop"
shaped. See if you can find out the name of the curve that is used, who first applied that
curve to roller coasters, and why that curve is used. At least one of the internet sources
gives some explanation...
Phase 6: The "Roller Coaster Scream"
Some background information:
•
Sound travels in pressure waves. These waves don't look like sine or cosine curves as
they travel through space, but if you plot pressure versus time (or distance) on a graph the
curve would look sinusoidal. Louder sounds involve more pressure (amplitude of the
sine curve), and higher pitched sounds have greater frequency (successive "peaks" are
closer together).
•
Waves move, so in describing waves it's necessary to talk about both time and distance
issues. One quantity relating time and distance is speed or velocity, and with a wave,
velocity refers to how fast a portion of the wave is moving. One way to think of this is to
imagine the speed that the crest of an ocean wave approaches the beach. The usual
rate*time=distance equation applies, even to waves! But waves have additional
properties: they have a periodic "shape." Two additional characteristics of waves are
frequency and wavelength. Frequency is concerned with time: how many waves pass a
fixed point in a given time. Wavelength is concerned with distance: how long is the
periodic shape? Frequency is denoted by various symbols, depending on the context; I'll
simply use f for frequency of a wave pattern, and the units for f are "waves per unit time"
(the unit Herz, or Hz, refers to waves per second). Wavelength is usually denoted by
λ (lambda), with units of "distance" (or "distance per wave"). f, λ, and velocity v are
related by the equation f λ=v, and a quick unit analysis confirms the logic in this
equation. An analogy for the frequency/ wavelength/ velocity relationship: Imagine
standing on the I-74 overpass on Prospect Avenue looking down at traffic, at 7:45 AM on
a week day. Traffic is moving at 65 mph (v); cars are fairly close together (λ small), and
lots of cars pass by in a short amount of time (f large). Now imagine standing on the
same overpass at 11:00 PM on a Sunday night. Traffic is still moving at 65 mph (v), but
now cars are few and far between: a very large λ (distance between cars), with a low f
(very few cars passing by in a fixed time).
•
The speed of sound in air is 348 m/sec (or 1140 ft/sec).
•
The human ear can detect pressure wave differences as small as 0.0002 microbars, or a
deviation of only 0.00000002 percent from normal atmospheric pressure!
•
(OPTIONAL) The doppler shift occurs when a wave-producing object and an observer
are moving toward or away from each other. You have probably heard a train whistle
drop in pitch suddenly as the train passed you. Think of what happens like this: as the
train moves toward you, its whistle emits a pressure burst (one piece of a sound wave),
which travels toward you at the speed of sound. By the time the whistle emits the next
burst (the next wave), the whistle has moved closer to you, so that second burst won't
have as far to travel as the first burst did. Since it's traveling at the same speed as the first
burst, it takes less time to reach you. So, if you measure "time between bursts" of the
sound you hear, your measurement is actually lower than the time between bursts at the
whistle. Since frequency is the reciprocal of "time between bursts," you hear a higher
frequency that the whistle is actually emitting. This is the doppler effect, and in the case
of a higher perceived frequency it's called a violet shift (violet light has the highest
frequency of the visible spectrum). If the train is moving away from you, however, each
burst takes longer to reach you than the preceding one, since it has further to travel. So
you measure "time between bursts" as longer than at the whistle, with a corresponding
decrease in the frequency of the sound you hear. This is known as a red shift. The
equation for the frequency a stationary observer hears is fO = c * fS /(c + v), where fO is
the frequency you Observe, c is the speed of sound, fS is the frequency at the Source, and
v is the velocity of the source, with a minus used for an approaching source and a plus
used for a receding source.
Your Requirements:
1. Those blood-curdling screams people on roller coasters tend to emit range in frequency
from about 800 Hz to upwards of 3500 Hz. Use the relationship among frequency,
wavelength, and velocity discussed in the background information to determine the range
in wavelengths for roller coaster screams.
2. An 80-decibel shriek results in a pressure wave with an amplitude of 2 microbars [source:
R.J. Baken, Clinical Measurement of Speech and Voice]. Recalling that frequency (waves per second) is the
reciprocal of period (seconds per wave), write an equation for p(t), the pressure in a
roller coaster scream, as a function of time (use a sinusoidal model). Use any frequency
you wish from the range given in item 1.
3. Make a careful graph of your p(t) from item 2. Be sure to label both axes, and show scale
clearly (obviously, your time scale should be fairly SMALL).
4. Write a paragraph telling why your roller coaster design will make riders SCREAM!
List all the "thrill" features of your roller coaster on the same sheet as your paragraph.
5. (OPTIONAL) Read the "optional" background information about the doppler shift. For
your group's roller coaster, assume an observer is standing at the bottom of the first hill,
when the train is traveling at its top speed (refer to your work for Phase 2). For the shriek
you analyzed in items 2 and 3, determine the frequency heard by the observer as the train
approaches, passes, and recedes from the observer.
Phase 7:The final design
"Specs" for the design contest (recap from the beginning of the project):
•
The roller coaster must fit on a 35 acre plot of land with no "holes" (you must count any
land lying between sections of track, or lying within loops, as part of the 35 acre plot),
BUT you may choose any dimensions for the plot.
•
The ride should last about two minutes.
•
The ride may have only one lift motor and one braking section.
•
The track may be metal or wood, but the "train cars" must be conventional (that is, they
must ride atop two parallel rails, and the rails are 6 feet apart [2 m. if you wish to use
metric units]).
•
The roller coaster ride must be on a track that is a closed circuit; that is, the train travels
along the track in one direction only, and returns to the station along the circuit to change
passengers.
•
The ride must be "safe."
Requirements
1. Make sure your designs from Phase 1 address the specifications above.
2. Write a summary proposal for the selection committee. Describe features of your
design, including type of material, safety features, "thrill factors," duration of the ride,
and any special features you think will make your design better than all the others. You
should also propose a name for your roller coaster. This proposal should be in the form
of a typed memorandum or letter, addressed "Dear Selection Committee," with money
stapled to the back for bribes; you may sign your proposal with "Sincerely, (your
'company name' here)." The quality of writing will be weighted equally with content in
the project grade. The length of a good proposal will be 1 (full!) to 2 pages, with a
maximum length of 3 pages. PLEASE DOUBLE-SPACE (most word processors allow
you to set spacing under "paragraph format").
3. Prepare a visual display to show how the name you chose for your roller coaster will
appear at the ride's entrance. Use color, but please, NO GLITTER. Your visual display
may be in the form of a poster (at least 18" square) or a computer graphic (powerpoint
slide, web page, shockwave file, etc.). If you use published material, including clip art,
you MUST acknowledge the source.
4. If there were deficiencies identified by the selection committee in any of the earlier
phases, you may make revisions or corrections. Jot a brief note (use a sticky note) to
alert the selection committee to re-check your earlier work.