Roller Coaster Physics
Intended for Grade:
Subject:
7th and 8th grade
Science and Math
Description: This project is designed to teach physical concepts such as velocity,
potential and kinetic energy, free fall motion, acceleration, and centripetal force by means
of a roller coaster. Students will be required to design and build a roller coaster using
these physical concepts which will be taught through various demonstrations and
experiments. The students will be introduced to basic economic principles so that they
may budget their money wisely in their design. In addition to this, students will create a
portfolio consisting of the design specifications, budget tables, and a news release all of
which will include a review of the concepts learned.
Mississippi State Framework addressed:
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7th grade Math: 3 c.- Use standard units of measurements to solve application
problems.
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7th grade Math: 8 c. – Determine unit rates.
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7th grade Science: 10 c. – Research and discuss energy transformation.
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7th grade Science: 10 d. – Convert one energy form to another.
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8th grade Math: 4c. – Solve proportions.
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8th grade Math: 6c. – Find the perimeter and area of polygons.
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8th grade Science: 10 c. – Research and discuss energy transformation.
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8th grade Science: 10 d. – Convert one energy form to another.
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General Social Studies (Economics): 1 d. – Describe how the laws of supply and
demand interact.
The project is primarily concerned with science and mathematics frameworks.
However, as a consequence, there will be some Language Arts frameworks covered.
National Framework addressed:
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Math Standard: Geometry
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Math Standard: Measurement
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Math Standard: Data Analysis
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Content Standard A: Science as Inquiry
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Content Standard B: Physical Science
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Lab 1: Velocity
Introduction
The purpose of this experiment is to explore the concept of velocity. This will be done
by means of a racecar traveling around a track. Velocity is a vector which means that it
not only is speed but speed with a direction. Therefore we can calculate velocity by
using distance/time but we must also give a direction like North, Northeast, South, etc.
Materials
2 photogates
1 electric racecar and track set
1 string
Calculation of velocity
1. Separate photogates 10cm on the racecar track.
2. Run the racecar around the track at a constant speed and record the time between
photogates for each distance.
3. Repeat the following experiment three times.
4. Find the average velocity.
5. Repeat the previous 4 steps for the photogates separated 20cm and 30cm.
6. Construct tables in your notebook that will hold all data and observations you’ve
found.
Conclusions
1. Will the velocity be greater or lesser as the distance between two photogates increase?
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2. If an object passes two photogates 10cm apart in a time of 0.50 seconds, what is the
velocity of the object? 20cm apart? 30cm?
3. Supposing the speed is constant, when on the track is the velocity constant?
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Lab 2: Galileo’s Inclined Plane
Introduction
This experiment is a reproduction of Galileo’s classic experiments with balls and inclined
planes. Before Galileo, people (including very smart people like Aristotle) believed that
falling objects’ acceleration was proportional to their mass. That is, they believed that
more massive (heavier) objects fell faster than less massive (lighter) objects. Your (the
students’) task in this activity is to test Aristotle’s hypothesis.
The equipment with which you have been provided closely resembles that available to
Galileo in the seventeenth century. He didn’t have photogate time sensors, so neither do
you. He didn’t have a stopwatch functional to hundredths of a second, so neither do you.
What Galileo had were an old-fashioned analog clock (the kind with a ticking second
hand), a meter stick, balls of different masses, and an assortment of boards and blocks.
This is the equipment with which Galileo performed his very famous experiment; it is
also the equipment with which you will now (300 years later) try to do the same.
Materials
Meter stick
Analog clock (Note: Stop watches can be used)
Balls of differing size and mass
Incline plane
Stands for incline plane
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Which Ball Rolls Faster?
This is the part of document where I usually tell you what to do. Sorry, not this time.
Galileo didn’t have instructions, and neither do you. You know what you have to do
(confirm or refute Aristotle’s hypothesis); figure out how to do it. Don’t worry, the
problem only stumped mankind for 1500 years. I’m sure you can handle it.
In a bound notebook, create a space to describe your actions in this activity. Write
everything down: time, date, location, coworkers’ names, procedures used, measurements
taken, and data. Any calculations that need doing should be done in this notebook.
Conclusions
Report, in paragraph form, what you did and how you did it. Show any calculations you
performed (complete with units, as always). Lastly, concisely state any conclusions you
were able to draw. When doing so, keep in mind that your work, reason, and imagination
should allow you to answer the following questions:
1. Do your results support or refute Aristotle’s hypothesis?
2. Why, physically, does Aristotle’s hypothesis fail/succeed?
3. Are there any conditions under which you might find a different result?
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Lab 3: Acceleration of a free falling body
Introduction
The objective of this experiment is to determine the acceleration of a freely
falling body due to the force of gravity. This will be accomplished by measuring the time
it takes for the body to fall specified distances.
The picket fence (Figure 1) will be used as the freely falling body.
Figure 1. Picket Fence
As you can see the picket fence is composed of alternating opaque and clear bands. This
will be important when it is dropped through the photogate because the opaque bands will
block the light (Figure 2).
Figure 2. This figure shows the setup
of the photogate and the picket fence
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The photogates will be used to measure the time between the breaks in light when the
picket fence is dropped. The distances between the bands can be measured. Using the
measured distances between the bands and the time between breaks in light, the velocity
can be calculated as in the velocity experiment.
Since the velocity of the picket fence increases as it falls, the change in velocity can be
calculated. Using the change in velocity during the time of the fall, the acceleration of
the fence can be calculated by using the following equation:
accelertation =
change in velocity
time for the change in velocity
Materials
1 picket fence
1 Photogate
1 Basket (trashcan)
1 Square piece of foam in basket
1 Metric ruler
Calculation of Acceleration
1. Set up the photogate and the picket fence as seen in Figure 2.
2. Drop the picket fence through the photogate letting the picket fence land on the
foam in the trash can.
3. Observe the velocity vs. time graph. You should notice that the velocity increases
at each successive opaque band. This means that the picket fence was
accelerating as it moved closer to the ground. Go to question #1.
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4. Now observe the acceleration vs. time graph. Write down the values the graph
gives for acceleration. Go to question #2.
Conclusions
1. Calculate the slope of your velocity vs. time graph (change in y divided by the
change in x). This value represents the acceleration of the picket fence. Indeed,
the change in y represents the change in velocity and the change in x represents
the change in time and acceleration is change in velocity over change in time.
2. Does your velocity vs. time graph support the concept that the acceleration of a
freely falling body is constant?
3. Find the average of your values for acceleration.
4. Does your acceleration vs. time graph support the concept that the acceleration of
a freely falling body is constant?
5. The true value for the acceleration of a freely falling body is 9.8 m/s2. How does
you average value and slope value compare to this?
6. Discuss possible sources of error in this experiment.
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Lab 4: Potential and Kinetic Energy and the
Conservation of Energy
Introduction
The purpose of this experiment is to introduce the concepts of potential and kinetic
energy by giving a ball a specific amount of potential energy and then letting the ball roll
down the Collider in order to calculate the kinetic energy. Conservation of Energy will
then be explored from these calculations.
When some object has energy, we mean that this object has the capacity to do work.
There are two types of energy to discuss; potential and kinetic energy. Potential energy
is energy due to position. When an object is held above ground, it has potential energy.
The object’s position (above ground) gives it this energy, because at this position the
object has the potential to do work. Once the object is ‘let go’, it starts working.
Potential energy (p.e.) is defined by
p.e. = (mass of object)x(acceleration due to gravity)x(height of object above ground)
or p.e.=mgh
where g=9.8 m/s2.
Kinetic Energy is energy in motion. Once an object, which has potential energy, is
released and starts moving, the potential energy then turns into kinetic energy. Therefore
kinetic energy involves velocity. Kinetic energy (k.e.) is defined by
k.e. = ½ x (mass of object)x(velocity squared)
or k.e.= ½ mv2 .
The law of conservation of energy states that when energy changes into a different form,
from potential into kinetic or kinetic into potential, none of it disappears and none is
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created. Basically whatever potential energy you started with will all be converted into
kinetic energy except for some that may have converted into heat due to friction.
Materials
2 photogates with timers
1 Collider (Note: of course this isn’t necessary, all that is needed is a steep curved incline
for the ball to roll down.)
1 ball
1 triple beam balance
Calculation of potential and kinetic energy
Note: To see good results in this experiment, measurements must be
accurate and the balls used to roll down the Collider should not be too small.
A lot of preparation is needed here.
1. Measure the mass of your ball in kilograms using the triple beam balance.
2. Set the photogates 10 cm apart on the inclined plane and record on the data sheet.
3. Determine the potential energy of your ball at this height.
4. Release the ball and record the velocity.
5. Repeat twice.
6. Determine the average value of velocity and record this number.
7. Determine average kinetic energy.
8. Construct tables in your notebook that will hold all data and observations you’ve
found.
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Conclusions
In your notebook make conclusions from data by answering the following questions:
1. Compare the potential energy with the kinetic energy. Discuss the conservation
of energy law and why potential and kinetic energy may not be exactly equal.
2. Now discuss these concepts has they occur on a roller coaster. Discuss potential
and kinetic energy as the cart starts from the top and rolls down and then rolls up
the track and then down again.
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Lab 5: Centripetal Force
Introduction
Things going around in circles experience special and interesting forces. We all know, for
instance, that a roller coaster train doesn't fall off the tracks as it goes through a loop.
Why is this? Is it simply because the train is hooked to the track? Or is there something
fundamental that holds the train on the track? If so, how fast does the train have to go to
not fall off? Does it matter? Does it matter how heavy the train is, or how big the loop
is?
This lab experiment will help you to answer all of these questions. Ultimately, what you
learn today will help you design a roller coaster that doesn't kill its passengers. So pay
attention, ask questions, and have fun!
FIG. 1. The electric rotor apparatus needed for this experiment. Note the basket, which
is moveable along the length of the rotor arm, and the dimmer switch, which allows for
fine control of the rotor speed.
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Procedure
The apparatus for this experiment has been constructed for you. It is expensive and
difficult to build; thus, you will need to
1. share it as a class, and
2. be careful not to damage it.
Note the important features of the apparatus. An electric motor turns a rotor (fan blade).
On this rotor is a series of small baskets, the openings of which face in toward the motor.
The baskets are placed at varying distances from the motor: 2.5cm, 5.0cm, 10.0cm, and
20.0cm. The motor itself is operated by a dimmer switch, which allows it to run at
varying speeds.
For the purposes of this experiment, the rotor apparatus will serve as a model roller
coaster loop. The marbles will be model trains. The basic format of the experiment is
simple: place a marble in the basket and turn on the rotor.
As we know, if we send a roller coaster train around a track fast enough, it won't fall off.
But the important question remains: why? To find out, we'll need to explore the
importance of several variables: mass of the train, the size of the loop, and the speed of
the train as it travels the loop. For the purposes of this experiment, the rotor apparatus
will serve as a model roller coaster loop. The marbles will be model trains. The basic
format of the experiment is simple: place a marble in the basket and turn on the rotor.
Observe the length and speed of the rotor, the mass of the marble, and whether or not the
marble falls “off the track,” i.e., out of the basket.
In order to evaluate the effects of each of these variables, we need to study them one at a
time. The following procedures will help you do exactly that.
Qualitative Experiments
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Rotor Speed
First, we will examine the effect of speed on “staying on track.” To begin, record your
hypothesis on the report sheet provided. Does speed matter? Will going faster make the
train more likely to stay on track, or less?
With your hypothesis recorded, we begin our experiment.
1. Pick a marble, any marble. Again, it doesn't matter which marble, but make sure
you use the same one throughout this part.
2. Place the marble in the 20.0cm basket.
3. Turn the rotor on as fast as it will go.
4. Keep reducing the speed of the rotor in small increments until the ball is just
barely staying in its basket. You should be able to hear a clicking noise as the ball
comes slightly away from the basket at the top of the rotor and falls back at the
bottom.
You've now determined the base speed the rotor needs to turn in order to keep the marble
in its basket. Now, we can examine the effects of marble mass and rotor length.
Marble Mass
Next, we will examine the effect of changing marble mass on “staying on track.” Again,
record your hypothesis before beginning. Will heavier marbles have to travel faster or
slower to stay in the basket?
1. Choose the least and most massive marbles.
2. Place the least massive marble in one 10.0 cm basket.
3. Place the most massive marble in the other 10.0 cm basket.
4. Spin up the fan and then reduce its speed gradually.
Examine the data for each marble. Describe the relationship between marble mass and
the speed needed to keep the marble in its basket. Does the heavier marble fall out before
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or after the lighter marble? When you build your coaster, will it matter how heavy the
train is?
Rotor Length
The last important factor we must discuss is rotor length. How will changing the size of
the loop affect the speed needed to keep the train on its track? We will determine this by
comparing the speeds needed to keep balls in baskets at different distances from the hub.
As always, record your hypothesis first. Will longer rotors have to move faster or slower
to keep the ball in its basket?
1. Place a marble in each basket.
2. Spin up the rotor.
3. Reduce the speed of the fan gradually.
In what order do the balls fall out of their baskets as the fan slows down? Use this
information to test your hypothesis: will your train have to move faster or slower around
a large loop than around a small loop?
WRAPUP
When you build a roller coaster, the important issue will be: How do I design the coaster
so that the train won't come off the track? At the same time, you'll want to consider: How
do I design the coaster so that the forces won't kill the passengers? When answering these
questions, the important issue is the speed of the coaster through any loops, corkscrews,
sharp turns, or other maneuvers. If the train is moving too slowly, it may fall off the
track while upside down. If it's moving too quickly, the force of the turn may mash the
passengers' brains into pudding.
The important quantity is centripetal force. As we've seen, the centripetal force (the force
holding the train to the track in a loop) depends on both the speed of the train and the
radius (size) of the loop or turn. The exact relationship, it turns out, looks like this:
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F = mv2/r
where $m$ is the mass of the train, v is the speed it's moving through the loop, and r is
the radius of the loop (or turn). Now, to keep our train on the tracks and our passengers
comfortable we need the centripetal force to fall somewhere between 10 m s-2 and 40 m s2
. As we’ve seen in our experiments, the mass of the train is not important. Thus, r and v
must be such that
10 m s-2 ≤ v2/r ≤ 40 m s-2
You can apply this last relationship to every loop in your coaster to ensure that the train
will stay on track and that the acceleration won't hurt the passengers. All you have to do
is square the speed of the train and divide by the radius of the loop. If the answer is
between 10 m s-2 and 40 m s-2, everything will be all right. If the force is greater than 40
m s-2, make the loop bigger. If it’s less than 10 m s-2, make the loop smaller.
Conclusions
Before we finish up, we need to think about a few things.
1. What will happen if your train moves too slowly through a loop?
2. What does this mean, in terms how high a loop should be relative to the hill
before it? Recall yesterday's lab before you answer.
3. What will happen if your train moves too fast through a loop?
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4. What does this mean, in terms how high a loop should be relative to the hill
before it? Recall yesterday's lab before you answer.
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The Budget
A local theme park wants to build a new roller coaster. The budget allotted for this
project is $21 million. Your mission is to design a roller coaster model within the
budgeting guidelines that all roller coaster contractors go by. These guidelines are as
follows:
1. The budget must not exceed $21 million.
2. You must allot 10% of your budget to emergencies and unforeseen costs during
construction.
3. You must allot the remaining 90% of the total budget to the building of the roller
coaster and landscape. This 90% allotted is what we call the actual cost of the
roller coaster.
4. The landscape and entrance to the roller coaster, what we call the Architecture,
are to be budgeted at 30% of the actual cost.
5. The roller coaster will be constructed with the remaining 70% of the actual cost.
6. You may use the notes and the attached worksheet to figure your budget. Once
you have worked the worksheet, ask your teacher if you can check your answers
with the Excel template titled “Roller Coaster Budget.”
Notes:
Roller coasters are generally one half mile in length. Researchers have found that riders
are not satisfied enough if the roller coaster is less than one half mile and tired of the ride
if it exceeds one half mile in length. Roller coaster contractors know that roller coasters
cost $1200 for each foot of track and $50 for each foot squared of area under the track.
Contractors always include “extra” track in their budget and plans to bring cars to and
from the actual running track. This “extra” track costs $1,000 for each foot, since it is
less complex than the actual riding track without area under the track. To figure out how
many feet you will need for your roller coaster, you must first figure out how many feet
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are in one mile and how much area you have under your track. You have your mission,
good luck!
FINDING THE AREA UNDER THE CURVE
In order for you to calculate the price of your roller coaster, you will need to find the area
under the tracks. Suppose the following curve represents the track of a roller coaster.
Finding the area under a curve is a technique in calculus called integration. We won’t
use integration but rather a simplified version of the idea behind it.
First, take your curve and place it on a grid and label your grid according to your scale.
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Next take your curve and divide it into familiar shapes whose area is easy to calculate.
Then sum together the areas of the familiar shapes. Note that this won’t be exact, but
will do for your budget, and you should try to make it as precise as you can.
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Portfolio
Each student will be responsible for their own personal notebook. However, groups
should be divided into four so that the students can work together. The worksheets from
each experiment, the design, the budget, and the newspaper article should all be within
the student’s portfolio. Each section is described in detail.
Worksheets
While completing each lab students should keep a worksheet of data. Some of the
activities lend themselves to charts. Theses should be completed using computer
resources. A clean and neat report should be included in the portfolio for each activity
completed. Several of the labs have conclusion questions listed. Each question should be
listed with a complete answer. If lab reports are used as handouts; students should
include original handouts within the portfolio.
Design
Using the concepts learned from the five previous experiments, students should begin
designing their roller coaster. Design specifications should work with the budget to
achieve the desired effects.
During the design process students are suggested to be creative when naming the coaster.
Encourage naming conventions that describe unique effects of the roller coaster.
Students should begin by searching for roller coasters on the internet. Important facts for
them to gather are: how tall is the roller coaster, how many feet the first drop has, what
kind of design the carts on the track have, etc. The Great American Scream Machine, the
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Mind Bender, the Batman ride, and the Superman Ultimate Flight Ride are good
examples. Each is unique in their design and students can easily find out why.
The students should now start sketching their roller coaster. Their sketch should be
scaled appropriately, maybe 1 foot per 1 thousand feet in length. With this scale though,
the height of the roller coaster is very small and so finding the area under the track would
be very difficult. In order to get a good sketch on a reasonably sized piece of paper, the
scale for the height of the coaster should be different than that of the length. This means
that when the students are calculating the area under the curve, they have two scales to
consider. If this seems difficult for the students, finding the area can be left out of the
design. Note that the roller coaster length can be measured with a string. All unique
features should be explicitly explained. Each sketch included in the portfolio should be
clean, clear, and descriptive.
In addition to this, the students can test their design on a roller coaster computer program.
Newspaper article
Students are required to write a newspaper article introducing their roller coaster to the
area. The article should include facts about the coaster such as cost, height, speed, and
physics concepts which were used in building the roller coaster. The article can be a
review or a promotional for the roller coaster. A three to four paragraph article is
suggested.
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Sources
1. The University of Mississippi, The department of Physics. Physical Science
laboratory experiments.
Prepared by:
Brian Hopkins
Carrie Darwin
Jaromy Kuhl
Leah Craft
Meredith Carnley
Michael Smith
NSF NMGK-8
University of Mississippi
July 2004
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