Acceleration in one, two, and three dimensions in launched roller

SPECIAL FEATURE: E XTREME P HYSICS
www.iop.org/journals/physed
Acceleration in one, two, and three
dimensions in launched roller
coasters
Ann-Marie Pendrill
Department of Physics, Göteborg University, SE 412 96 Göteborg, Sweden
E-mail: [email protected]
Abstract
During a roller coaster ride, the body experiences acceleration in three
dimensions. An accelerometer can measure and provide a graph of the forces
on the body during different parts of a ride. To couple the experience of the
body to pictures of the ride and an analysis of data can contribute to a deeper
understanding of Newton’s laws. This article considers the physics of
launched roller coasters. Measurements were performed with a
three-dimensional co-moving accelerometer. An analysis is presented of the
forces in the different ride elements of the Kanonen in Göteborg and the
Speed Monster in Oslo, which both include loops and offer rich examples of
force and acceleration in all dimensions.
Introduction
3, 2, 1 . . . launch! The traditional lift hill, which
gives the initial potential energy for the ride, is
absent in some newly built roller coasters. Instead,
the initial energy is provided in the form of a
horizontal launch, giving sufficient kinetic energy
to bring the train to the top of the first hill.
From then on, the ride is characterized by the
interchange between potential and kinetic energy,
in the same way as in traditional roller coasters.
The first Intamin hydraulic launch coaster in
Europe was Rita the Ride at Alton Towers, which
opened in April 2005, followed two weeks later by
Kanonen at Liseberg in Göteborg (figure 1). The
Speed Monster at Tusenfryd in Oslo (figure 2) and
the Stealth at Thorpe Park (figure 3) both opened
in 2006. In 2007, similar launch coasters were
added to Heide-Park in Germany and PortAventura
in Spain [1–3]. (See also the Roller Coaster Data
Base at www.rcdb.com.)
The Stealth is the highest of the European
launch coasters. After the launch, the train passes
the ‘top hat’ (figure 3) and then returns over a
camel back into a hairpin turn back into the station.
The Kanonen and Speed Monster roller
coasters both feature a loop and a screw during the
ride. The accelerometer data and elevation profile
from these rides are shown in figures 4 and 5, and
discussed in more detail below.
One-dimensional horizontal motion
In schools, the study of motion traditionally starts
with non-motion, continuing with motion in one
dimension. The traditional lift hill is an example
of uniform rectilinear motion, where Newton’s
first law applies. The launch is an example of
accelerated motion in one dimension—as is the
final brake. These situations can be useful as
illustrations to textbook presentations. In one
dimension, the measurement of the acceleration
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A-M Pendrill
Figure 1. The Kanonen roller coaster viewed from the side, showing the launch from the left into the ‘top hat’ on
the right, as well as the shape of the clothoid loop.
Figure 2. Panorama of the Speed Monster. The launch is from the right into the Norwegian loop, which encircles
the entrance escalator. (Photo: Jochen Peschel [1].)
in the direction of motion gives full information
about the motions if the initial speed is known.
The variation of speed and distance with time is
obtained by integration, which can be performed
numerically or analytically, after approximation of
the acceleration time dependence.
The launch
Flags in the launch area enhance the sensation
of motion during launch of the Speed Monster,
as shown in figure 6. Horizontal launches of
roller coasters have been used since the 1970s:
for example, in the Revolution [2] which is a
Schwarzkopf ‘shuttle launch coaster’ [4], where
the energy is stored in a flywheel. Magnetic launch
techniques were introduced during the 1990s, with
LIMs (linear induction motors) and LSMs (linear
synchronous motors). The compressed air launch
was introduced in 2002, followed by the hydraulic
launch in 2002. The hydraulic launch was used
to break a new altitude record in 2003 for the Top
Thrill Dragster at Cedar Point, Ohio [2, 5].
In the hydraulic launch, oil is pumped
from a reservoir into storage cylinders filled
with nitrogen. The energy is built up as the
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PHYSICS EDUCATION
nitrogen is compressed to a pressure of around
300 bar. During launch, the gas is allowed to
expand rapidly, sending the hydraulic oil through
the motors, and energy is transferred to the
accelerating roller coaster. The technique is
described in some detail by Peschel [3], who
also presents an animation of the launch process.
The pressure drops to about 250 bars, consistent
with the drop in horizontal acceleration during the
launch, seen from the graphs in figures 7 and 8.
Figures 7 and 8 shows the accelerometer
data for the Kanonen and Speed Monster rides.
The graphs also include speed and distance,
obtained by numerical integration. From the
graphs in figures 7 and 8, we can conclude that
the force drops during the launch. This is natural
since the pressure of the nitrogen would drop as
the gas expands, as discussed below. A fully
loaded Kanonen train with four cars weighs about
8 tonnes. The Speed Monster train with three cars
is lighter, about 6 tonnes. These weights include
the mass of the sled used during acceleration.
Exercises for the reader.
• What average power is needed to accelerate
the trains (in W and horsepower
(1 hp = 735 W))?
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Acceleration in one, two, and three dimensions in launched roller coasters
avert/g
4
2
0
height (m)
0
5
10
15
20 25
t(s)
30
35
40
45
20
10
0
0
5
10
15
20
25
30
35
t(s)
Figure 5. Accelerometer and elevation data for the
Speed Monster.
avert/g, atot/g
Figure 3. The 62 m high ‘top hat’ of the Stealth roller
coaster at Thorpe Park.
Figure 6. The launch of the Speed Monster, with a side
view of the ‘Norwegian loop’.
4
2
0
–2
height (m)
0
5
10
15
t(s)
20
25
30
35
• How high above the starting point can the
Kanonen and Speed Monster trains go after
launch?
20
10
Acceleration measurements in three
dimensions
0
0
5
10
15
20
25
30
35
t(s)
Figure 4. Accelerometer and elevation data for
Kanonen. The green accelerometer curve shows
magnitude of the ‘g-force’, whereas the blue curve
shows only the vertical component.
• How does the power, P , vary during the
launch? (Remember that power is force times
velocity, P = Fv .)
September 2008
In roller coasters, as in everyday life, acceleration
is rarely restricted to one dimension. The forces
required for the acceleration in a roller coaster are
evident throughout the body. What the body can
experience can also be measured with a co-moving
sensor. Since the body moves in the gravitational
field, g, from the Earth, the additional force per
mass unit required to obtain an acceleration, a, is
(a − g). What is measured by an accelerometer is
thus in general not acceleration, but one or more
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485
30
20
10
0
0
1
2
2
Figure 7. Horizontal acceleration (m/s ) for the
Kanonen launch, together with velocity (m/s) and
distance (m) obtained through numerical integration.
Which graph is which? The drop in acceleration in
figures 7 and 8 corresponds to a drop in force and
thus to the drop in pressure during launch, which can
be used to estimate the fraction of the maximum
possible work exerted by the gas during the Kanonen
and Speed Monster launches.
components of this vector. Since the gravitational
acceleration is used as a reference, it is natural to
give results in terms of the ratio (a − g)/g . This
expression can be taken as a vector definition of
the ‘g -force’.
The accelerometer data in this paper were
obtained using a wireless dynamic sensor system
from Vernier. This system also measures the
air pressure and converts the barometer data to
provide indications of altitude during the ride.
Through Bernoulli’s principle, the altitude data
are influenced by speed, thus leading to an
overestimate of altitude for high speeds. (This can
be seen, for example, around launch in the graphs
in figures 4 and 5.)
Coordinate system for amusement ride
acceleration data
The experience of the body depends on the
orientation.
A natural coordinate system to
describe the experience follows the moving body,
thus changing direction throughout the ride, and
this is also the coordinate system used by the
sensor to record the motion. Here, we define the
positive z -axis to be the ‘vertical’ axis directed
along the spine towards the head of the rider. The
positive x -axis points to the front of the rider—in
most roller coaster rides, including these ones, the
x -axis coincides with the direction of motion. The
y -axis gives the direction of the ‘lateral’ g -force.
In a right-handed system it will point out to the
left of the rider. Apart from launch and brake, the
longitudinal component should vanish if friction
and train length are neglected. Except for screw
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x(m), vx(m/s), ax(m/s2)
A-M Pendrill
30
20
10
0
2
3
4
5
t(s)
Figure 8. Launch of the Speed Monster: acceleration
2
(m/s ), velocity (m/s) and distance (s).
elements in a roller coaster, the lateral components
vanish if the curves are perfectly banked.
A problem in measuring acceleration in three
dimensions is to keep the sensor axis aligned
with the body axis. When the sensor is kept
safe in a vest on the body, the z -axis tends
to slope slightly backwards and sometimes also
sideways. A mathematically simple option is to
use the magnitude of the vector |a − g|, possibly
incorporating the sign from the dominating
vertical component to maintain ‘negative g ’
readings. The Kanonen data in figure 4 show a
comparison of the total g -force and the vertical
component.
When the other coordinates are also of
interest, as for launch, break and roll, it is
necessary to perform a coordinate transformation.
The data in this paper were transformed by rotating
the axes so that the data have only a vertical
component before the ride starts, and assuming
that the sensor orientation relative to the track is
fixed.
Two-dimensional motion in loops
Both the Kanonen and the Speed Monster include
loops, where the train moves essentially in two
dimensions. The photographs in figures 1 and 9
show that neither the loop in the Kanonen nor
in the Speed Monster is a perfect circle. In a
circular loop, weightlessness at the top would be
accompanied by 6g at the bottom of the loop
(neglecting energy losses and the length of the
train). To reduce the load on the body, the shape
of the track has a larger radius of curvature at
the bottom. This can be achieved in different
ways, as discussed in more detail in [6, 7]. The
Kanonen loop is a classic ‘clothoid loop’, which
was introduced by Werner Stengel in 1976 in the
roller coaster revolution [6].
September 2008
Acceleration in one, two, and three dimensions in launched roller coasters
Figure 9. The ‘Norwegian loop’ of the Speed Monster
roller coaster encircles the roller coaster entrance to the
park, making a very large loop possible. In view of the
short Speed Monster train, the ratio between train
length and loop radius thus becomes unusually small in
this case.
In traditional roller coaster loops, the train
enters the loop from below. The Speed Monster
train instead enters the loop from above. This
feature, conceived by project director Morten
Bjerke at Tusenfryd, makes the Speed Monster
loop unique. Is is classified as a ‘Norwegian
loop’ in the Roller Coaster Data Base (www.rcdb.
com). It gives the rider two inversions, during both
entrance to and exit from the loop.
The Kanonen train passes the highest point at
time 15 s in the data series in figure 4, showing
essential weightlessness at the top and close to
4g during entrance to and exit from the loop.
Similarly, the Speed Monster rider is essentially
weightless at the entrance and exit from the loop
(at 10 s and 15 s, respectively, in figure 5), while
experiencing close to 4.5g at the bottom.
Comparing the loop shapes, we see that,
whereas the traditional loops are somewhat
narrower than a circle, the larger curvature at the
bottom of the Norwegian loop leads instead to a
slightly wider shape.
Three-dimensional motion in corkscrews
The picture of the Speed Monster launch (figure 6)
also shows the large Norwegian loop from the
side. All of the loop is nearly in the same plane.
Separating the coils by a larger distance would lead
to a corkscrew, such as in the Speed Monster, as
seen in figures 2 and 10.
A corkscrew can, as a first approximation,
be described in cylindrical coordinates, where
the circular motion with a radius R is then
September 2008
Figure 10. The large corkscrew of the Speed Monster.
Technically, only the last hill is considered as a
corkscrew element. The track twists so that the train
runs on top of the track in the first two coils. Only the
last coil leads to an inversion of the rider. The train
position in the photo corresponds to t = 33 s in the
graph in figure 5.
accompanied by a perpendicular motion along the
cylinder axis. In the photograph of the corkscrew
in figure 2, the track seems a bit flattened at the top.
At the same time, the track twists, so the heartline
of the rider moves more along the cylindrical
shape. Let L denote the distance between the coils
along the axis. For the train to move a full coil, it
then moves a distance 2π R around the circle and
L along the axis. The velocity component along
the cylinder axis is unchanged during the motion.
The angle of the track to the axis is given by
tan α = 2π R/L.
Exercise.
• Show that a train moving with speed v along
a corkscrew track leads to a centripetal
acceleration with magnitude
(v sin α)2
v2
1
ac =
=
.
R
R 1 + L 2 /4π 2 R 2
• Show that the difference between the g -force
at the top and bottom of the corkscrew is
given by
4
2g + 4g sin2 α = g 2 +
.
1 + L 2 /4π 2 R 2
In the formula above, the first 2g arise due
to the different direction of the body relative to
gravity, when the rider stays on the inside of
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A-M Pendrill
Kanonen, Heartline Roll
a/g
2
total
0
lateral
–2
25
Figure 11. The photograph shows the Kanonen train on
the way back through the loop into the heartline roll,
where the centre of mass of the rider moves essentially
along a straight line.
the screw and is upside down at the top. This
obviously does not apply in situations, such as
figure 12, where the train has twisted around to the
top of the track in the highest point. Corrections
may also arise due to the motion of the train around
the track. Any difference in radius of curvature
between the high and low points also leads to a
change in g -force difference to what is expected
from these formulae.
Speed Monster corkscrew
The corkscrew in the Speed Monster is quite
stretched, making good use of the available space,
as seen from the panorama picture in figure 2. In
the Kanonen ride, the corkscrew is stretched to the
point where the riders move along a straight line
while the track twists around them, giving a ratio
R/L close to zero.
As an exercise, estimate R/L for the Speed
Monster corkscrew from figure 2. Use this
ratio to estimate the difference in g -force for the
different parts of the ride. Does your result agree
with the accelerometer data in figure 5, where
the corkscrew spans the period of about 10 s,
starting at t = 28 s? Are there any deviations
from expectations that would prompt you to make
additional observations or measurements in the
park?
The heartline roll of Kanonen
On the way back to the station, the Kanonen train
performs a show-off passage over guests in the
queue (figure 11). The track turns about 270◦ in a
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26
vertical
27
28
29
t(s)
30
31
32
33
Figure 12. Accelerometer data for the ‘heartline roll’.
The vertical (red) and lateral (green) components are
shown together with the total g-force (black) on the
body. Since the body moves with essentially constant
velocity, the total force from the train on the body is
mg, counteracting the force of gravity throughout the
roll. However, the direction is changed relative to the
rotating coordinate system of the body and of the
accelerometer.
‘heartline roll’. The body’s centre of mass moves
with nearly constant velocity. What forces act on
the body? Figure 12 shows the accelerometer data
for this part of the tour.
It is tempting to believe that measurement
with a three-dimensional accelerometer gives a
complete description of the motion, which can be
used to recreate the shape of the track. However,
the accelerometer data between 27 and 30 s in
figure 12 could be obtained without rotation by
moving up–down and left–right, some 20 m in
each direction (although the altitude profile does,
indeed, show that this was not the case).
Newton’s first law tells us that a body remains
in uniform rectilinear motion unless acted on by
unbalanced forces. However, when the ‘body’
in Newton’s laws is our own it is clear that
the direction of the forces relative to the body
matters: we are not point-like particles. A ‘motion
tracker’ needs also to measure rotation around
the three axes to get a complete description of
the motion [9]. Nevertheless, three-dimensional
accelerometer data provide much material for
analysing familiar motions.
Personal experiences
The wireless sensor system
The WDSS system is extremely simple to use.
Once set up from a computer, it can be used by
a large number of students to collect data over
a whole afternoon. It is, however, important to
keep notes of what rides have been studied, since
no additional information is stored on the sensor.
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Acceleration in one, two, and three dimensions in launched roller coasters
Should the memory fill up, the data are quickly
transferred to a laptop, and the sensor is again
ready for additional measurements.
The measuring vest, although not particularly
aesthetically pleasing, seems to convince ride
attendants that the wearer is serious.
Most
importantly, the vest keeps the sensor from falling
out during the ride: safety concerns must always
come first. A disadvantage is, however, that it is
difficult to keep the coordinate axes aligned [10]1 ,
but in most cases this can be dealt with afterwards.
One-dimensional accelerometer data are sufficient to obtain velocity and position rectilinear
motion, at least in principle. For a complete description of three-dimensional motion, accelerometer data for the three axes must be complemented
by rotational data around all axes, as discussed by
Pendrill and Rödjegård [9], in connection with the
analysis of motion tracker data for a roller coaster.
Still, the simplicity of use for the WDSS sensor
makes it a useful tool, bringing the amusement
park experience to the classroom.
are, of course, during off-season, when you do not
have to wait more than a few minutes to board the
train.
Rita—Queen of Speed, Kanonen and the
Speed Monster all have slower speeds and lower
hills than the Stealth. Rita—Queen of Speed
reaches a higher speed (98 km h−1 ) than the
Scandinavian launch coasters, but the ride heights
at Alton Towers are limited by the tree tops. The
speed gained from the launch is instead used in a
helix with an extended period of relatively strong
g -forces. The whole 640 m tour in Rita the Ride
lasts 25 s, which may seem a bit short after a
long time in the queue. Alton Towers has more
rewarding roller coaster rides!
The Kanonen and the Speed Monster both
turn the rider upside down a few times during
the ride, in loops and screws, discussed above.
Although the inversions could be captured by a
rotational sensor, the visual experience could not.
The Kanonen launch goes across a small river,
giving the riders the impression of falling into
the water after the top hat. The Speed Monster
has a most spectacular track layout, encircling the
The roller coasters
Although a three-dimensional accelerometer records entrance escalators. It runs on a hillside, and
the time series of forces acting on the body, it can brings the rider through the terrain, close to the
tree tops. The Kanonen track is woven back and
obviously not capture the whole experience.
Part of the experience is the build-up of forth, making maximum use of a small available
expectations during the time in the queue. The area. Its complicated structure is more difficult to
Stealth queue at Thorpe offers TV screens with memorize, which possibly brings more surprises
its own disc jockey. During my one-hour wait to to the rider. The Speed Monster makes use of the
get on (August 2006), people in the queue were natural drops to bring the train considerably below
dancing to the music, and in general having a the starting point, thereby increasing the maximum
good time. There was also the occasional speaker speed. The ride is only about 2 s longer than
message telling NN to get ‘back to the entrance the Kanonen ride, as seen from the accelerometer
where mum has got a FastTrack ticket for you’. data. However, the difference feels larger, possibly
Just before entering, the riders are brought in close because the Speed Monster track is more than 50%
view of parts of the launch technology. The queue longer: 690 m compared to 440 m. (The Stealth
also lets you look up towards the 62 m high top tour is even shorter: 400 m.) The longer track
hat (figure 3), and I have to confess that it was also accounts for the smoother ride, where more
the first time in many years that I had the feeling distance is allowed for the different elements [8].
Which coaster is the ‘best’? To some extent
‘am I really going to go on that ride?’. But, yes,
I did, and even got a FastTrack ticket for a second this depends on your personal preferences. The
ride. The long period of acceleration followed by results from annual voting by riders can be found
at BestRollerCoasterPoll.com.
weightlessness is quite a strong experience.
Both the Kanonen and the Speed Monster
offer good views of different parts of the ride as Lessons in the amusement park or roller coasters
the queue moves on. The best queuing experiences in the classroom?
Is it best to have physics lessons in the amusement
1 Figure 3 of [10] shows the spillover from the vertical
park or to have lessons in school about physics
accelerometer reading, which results when the coordinate axes
are not correctly aligned.
in amusement rides? Even without easy access
September 2008
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A-M Pendrill
to an amusement park, most students are likely
to have been on the rides and can relate the
experience of their body to the physics description
of the rides. Swings in a nearby playground
are a good way to introduce amusement park
physics [11]. Discussions of forces in the rides
are likely to change students’ ways of thinking
during future park visits, as many students have
reported. As with all field trips [12], the learning
outcome from an amusement park visit depends
to a large extent on the preparation. Several
Internet sites provide material for preparing
amusement park visits (e.g. [13, 14]). Some
parks, including Alton Towers and Thorpe Park,
offer educational programmes for visiting school
classes.
Thorpe Park also quotes education
secretary Alan Johnson: ‘Learning outside the
classroom should be at the heart of schools’
curriculums and ethos.’
Measurements in the park can easily overshadow analysis, which is left for later. Back
in the classroom, the rides are no longer at hand
for investigating questions arising from the data.
The balance between measurement and analysis is
worth careful consideration. The analysis of measurement data also takes somewhat different forms
depending on what data can be obtained from the
park. Drawings are usually secret, as required by
the agreements between parks and designers. The
length of a roller coaster train can, however, usually be obtained. (If not, it can be estimated by
measuring the width of the gates in the boarding
queues.) It can provide a length scale for analysis of different elements of the roller coaster from
photographs or video clips. The length, combined
with the time of passage at a given point, gives a
speed measurement. Comparing timing from stop
watches of a number of students’ mobile phones
provides good material for discussions of measurement uncertainty. Sometimes the track layout also
makes it possible to estimate energy losses from
measurement of the time of passage.
I find that every time I get new data from a
ride, they give rise to questions, and an urge to go
back and check. Now, I would like to go back to
Tusenfryd and take a good look at the corkscrew
to see if what looks like an extra large radius of
curvature at the bottom of one of the coils can
explain the dip in g -force around 35 s in figure 5;
and I would need to ride it again to feel that dip.
I would also like to feel the ‘negative g -force’ at
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PHYSICS EDUCATION
the top of the first coil (figure 6, at about 28 s in
the data shown in figure 5). Measurements are not
only about numbers, but about questions, answers
and insight.
Acknowledgments
First, I would like to express my appreciation
to roller coaster designer Werner Stengel for
kindly sharing part of his knowledge about various
aspects of roller coasters, including loop shapes.
I would also like to thank Jochen Peschel from
Coasters and More for permission to use the
photograph in figure 2, and for interesting e-mail
correspondence. Finally, I would like to thank
the helpful people at Liseberg and Tusenfryd, in
particular Ulf Johansson and Morten Bjerke, for
practical help and for stimulating discussions.
Received 2 January 2008, in final form 6 February 2008
doi:10.1088/0031-9120/43/5/003
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Acceleration in one, two, and three dimensions in launched roller coasters
[12] Rennie L R and McClafferty T P 1996 Science
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LISEBERG/
September 2008
Ann-Marie Pendrill is professor in
physics at Göteborg University, with a
background in computational atomic
physics. Her teaching involves
engineering, physics and teacher
programmes and she is involved with
different forms of informal learning,
including amusement park physics.
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