SP EC I A L F E A T UR E : E X T R EM E P HY S I C S

www.iop.org/journals/physed

Acceleration in one, two, and threedimensions in launched rollercoastersAnn-Marie Pendrill

Department of Physics, Goteborg University, SE 412 96 Goteborg, Sweden

E-mail: Ann-Marie.Pendrill@physics.gu.se

AbstractDuring a roller coaster ride, the body experiences acceleration in threedimensions. An accelerometer can measure and provide a graph of the forceson the body during different parts of a ride. To couple the experience of thebody to pictures of the ride and an analysis of data can contribute to a deeperunderstanding of Newton’s laws. This article considers the physics oflaunched roller coasters. Measurements were performed with athree-dimensional co-moving accelerometer. An analysis is presented of theforces in the different ride elements of the Kanonen in Goteborg and theSpeed Monster in Oslo, which both include loops and offer rich examples offorce and acceleration in all dimensions.

Introduction3, 2, 1 . . . launch! The traditional lift hill, whichgives the initial potential energy for the ride, isabsent in some newly built roller coasters. Instead,the initial energy is provided in the form of ahorizontal launch, giving sufficient kinetic energyto bring the train to the top of the first hill.From then on, the ride is characterized by theinterchange between potential and kinetic energy,in the same way as in traditional roller coasters.The first Intamin hydraulic launch coaster inEurope was Rita the Ride at Alton Towers, whichopened in April 2005, followed two weeks later byKanonen at Liseberg in Goteborg (figure 1). TheSpeed Monster at Tusenfryd in Oslo (figure 2) andthe Stealth at Thorpe Park (figure 3) both openedin 2006. In 2007, similar launch coasters wereadded to Heide-Park in Germany and PortAventurain Spain [1–3]. (See also the Roller Coaster DataBase at www.rcdb.com.)

The Stealth is the highest of the Europeanlaunch coasters. After the launch, the train passesthe ‘top hat’ (figure 3) and then returns over acamel back into a hairpin turn back into the station.

The Kanonen and Speed Monster rollercoasters both feature a loop and a screw during theride. The accelerometer data and elevation profilefrom these rides are shown in figures 4 and 5, anddiscussed in more detail below.

One-dimensional horizontal motionIn schools, the study of motion traditionally startswith non-motion, continuing with motion in onedimension. The traditional lift hill is an exampleof uniform rectilinear motion, where Newton’sfirst law applies. The launch is an example ofaccelerated motion in one dimension—as is thefinal brake. These situations can be useful asillustrations to textbook presentations. In onedimension, the measurement of the acceleration

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Figure 1. The Kanonen roller coaster viewed from the side, showing the launch from the left into the ‘top hat’ onthe 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 encirclesthe entrance escalator. (Photo: Jochen Peschel [1].)

in the direction of motion gives full informationabout the motions if the initial speed is known.The variation of speed and distance with time isobtained by integration, which can be performednumerically or analytically, after approximation ofthe acceleration time dependence.

The launch

Flags in the launch area enhance the sensationof motion during launch of the Speed Monster,as shown in figure 6. Horizontal launches ofroller coasters have been used since the 1970s:for example, in the Revolution [2] which is aSchwarzkopf ‘shuttle launch coaster’ [4], wherethe energy is stored in a flywheel. Magnetic launchtechniques were introduced during the 1990s, withLIMs (linear induction motors) and LSMs (linearsynchronous motors). The compressed air launchwas introduced in 2002, followed by the hydrauliclaunch in 2002. The hydraulic launch was usedto break a new altitude record in 2003 for the TopThrill Dragster at Cedar Point, Ohio [2, 5].

In the hydraulic launch, oil is pumpedfrom a reservoir into storage cylinders filledwith nitrogen. The energy is built up as the

nitrogen is compressed to a pressure of around300 bar. During launch, the gas is allowed toexpand rapidly, sending the hydraulic oil throughthe motors, and energy is transferred to theaccelerating roller coaster. The technique isdescribed in some detail by Peschel [3], whoalso presents an animation of the launch process.The pressure drops to about 250 bars, consistentwith the drop in horizontal acceleration during thelaunch, seen from the graphs in figures 7 and 8.

Figures 7 and 8 shows the accelerometerdata for the Kanonen and Speed Monster rides.The graphs also include speed and distance,obtained by numerical integration. From thegraphs in figures 7 and 8, we can conclude thatthe force drops during the launch. This is naturalsince the pressure of the nitrogen would drop asthe gas expands, as discussed below. A fullyloaded Kanonen train with four cars weighs about8 tonnes. The Speed Monster train with three carsis lighter, about 6 tonnes. These weights includethe mass of the sled used during acceleration.

Exercises for the reader.

• What average power is needed to acceleratethe trains (in W and horsepower(1 hp = 735 W))?

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Figure 3. The 62 m high ‘top hat’ of the Stealth rollercoaster at Thorpe Park.

4

2

0

–2

a ver

t/g, a

tot/g

25 30 3520151050

20

0heig

ht (

m)

25 30 3520151050t(s)

10

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 thelaunch? (Remember that power is force timesvelocity, P = Fv.)

4

2

0

a ver

t/g

35 40 4525201550

20

0heig

ht (

m)

25 30 3520151050t(s)

10

t(s)

Figure 5. Accelerometer and elevation data for the Speed Monster.

3010

Figure 6. The launch of the Speed Monster, with a sideview of the ‘Norwegian loop’.

• How high above the starting point can theKanonen and Speed Monster trains go afterlaunch?

Acceleration measurements in threedimensionsIn roller coasters, as in everyday life, accelerationis rarely restricted to one dimension. The forcesrequired for the acceleration in a roller coaster areevident throughout the body. What the body canexperience can also be measured with a co-movingsensor. Since the body moves in the gravitationalfield, g, from the Earth, the additional force permass unit required to obtain an acceleration, a, is(a − g). What is measured by an accelerometer isthus in general not acceleration, but one or more

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20

10

0

210

Figure 7. Horizontal acceleration (m/s2) 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.

30

components of this vector. Since the gravitationalacceleration is used as a reference, it is natural togive results in terms of the ratio (a − g)/g. Thisexpression can be taken as a vector definition ofthe ‘g-force’.

The accelerometer data in this paper wereobtained using a wireless dynamic sensor systemfrom Vernier. This system also measures theair pressure and converts the barometer data toprovide indications of altitude during the ride.Through Bernoulli’s principle, the altitude dataare influenced by speed, thus leading to anoverestimate of altitude for high speeds. (This canbe seen, for example, around launch in the graphsin figures 4 and 5.)

Coordinate system for amusement rideacceleration data

The experience of the body depends on theorientation. A natural coordinate system todescribe the experience follows the moving body,thus changing direction throughout the ride, andthis is also the coordinate system used by thesensor to record the motion. Here, we define thepositive z-axis to be the ‘vertical’ axis directedalong the spine towards the head of the rider. Thepositive x-axis points to the front of the rider—inmost roller coaster rides, including these ones, thex-axis coincides with the direction of motion. They-axis gives the direction of the ‘lateral’ g-force.In a right-handed system it will point out to theleft of the rider. Apart from launch and brake, thelongitudinal component should vanish if frictionand train length are neglected. Except for screw

20

10

0

532

Figure 8. Launch of the Speed Monster: acceleration (m/s2), velocity (m/s) and distance (s).

30

4x(m

), v

x(m

/s),

ax(

m/s

2 )

t(s)

elements in a roller coaster, the lateral componentsvanish if the curves are perfectly banked.

A problem in measuring acceleration in threedimensions is to keep the sensor axis alignedwith the body axis. When the sensor is keptsafe in a vest on the body, the z-axis tendsto slope slightly backwards and sometimes alsosideways. A mathematically simple option is touse the magnitude of the vector |a − g|, possiblyincorporating the sign from the dominatingvertical component to maintain ‘negative g’readings. The Kanonen data in figure 4 show acomparison of the total g-force and the verticalcomponent.

When the other coordinates are also ofinterest, as for launch, break and roll, it isnecessary to perform a coordinate transformation.The data in this paper were transformed by rotatingthe axes so that the data have only a verticalcomponent before the ride starts, and assumingthat the sensor orientation relative to the track isfixed.

Two-dimensional motion in loopsBoth the Kanonen and the Speed Monster includeloops, where the train moves essentially in twodimensions. The photographs in figures 1 and 9show that neither the loop in the Kanonen norin the Speed Monster is a perfect circle. In acircular loop, weightlessness at the top would beaccompanied by 6g at the bottom of the loop(neglecting energy losses and the length of thetrain). To reduce the load on the body, the shapeof the track has a larger radius of curvature atthe bottom. This can be achieved in differentways, as discussed in more detail in [6, 7]. TheKanonen loop is a classic ‘clothoid loop’, whichwas introduced by Werner Stengel in 1976 in theroller coaster revolution [6].

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Figure 9. The ‘Norwegian loop’ of the Speed Monsterroller coaster encircles the roller coaster entrance to thepark, making a very large loop possible. In view of theshort Speed Monster train, the ratio between trainlength and loop radius thus becomes unusually small inthis case.

In traditional roller coaster loops, the trainenters the loop from below. The Speed Monstertrain instead enters the loop from above. Thisfeature, conceived by project director MortenBjerke at Tusenfryd, makes the Speed Monsterloop unique. Is is classified as a ‘Norwegianloop’ in the Roller Coaster Data Base (www.rcdb.com). It gives the rider two inversions, during bothentrance to and exit from the loop.

The Kanonen train passes the highest point attime 15 s in the data series in figure 4, showingessential weightlessness at the top and close to4g during entrance to and exit from the loop.Similarly, the Speed Monster rider is essentiallyweightless at the entrance and exit from the loop(at 10 s and 15 s, respectively, in figure 5), whileexperiencing close to 4.5g at the bottom.

Comparing the loop shapes, we see that,whereas the traditional loops are somewhatnarrower than a circle, the larger curvature at thebottom of the Norwegian loop leads instead to aslightly wider shape.

Three-dimensional motion in corkscrewsThe picture of the Speed Monster launch (figure 6)also shows the large Norwegian loop from theside. All of the loop is nearly in the same plane.Separating the coils by a larger distance would leadto a corkscrew, such as in the Speed Monster, asseen in figures 2 and 10.

A corkscrew can, as a first approximation,be described in cylindrical coordinates, wherethe circular motion with a radius R is then

Figure 10. The large corkscrew of the Speed Monster.Technically, only the last hill is considered as acorkscrew element. The track twists so that the trainruns on top of the track in the first two coils. Only thelast coil leads to an inversion of the rider. The trainposition in the photo corresponds to t = 33 s in thegraph in figure 5.

accompanied by a perpendicular motion along thecylinder axis. In the photograph of the corkscrewin figure 2, the track seems a bit flattened at the top.At the same time, the track twists, so the heartlineof the rider moves more along the cylindricalshape. Let L denote the distance between the coilsalong the axis. For the train to move a full coil, itthen moves a distance 2π R around the circle andL along the axis. The velocity component alongthe 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 alonga corkscrew track leads to a centripetalacceleration with magnitude

ac = (v sin α)2

R= v2

R

(1

1 + L2/4π2 R2

).

• Show that the difference between the g-forceat the top and bottom of the corkscrew isgiven by

2g + 4g sin2 α = g

(2 + 4

1 + L2/4π2 R2

).

In the formula above, the first 2g arise dueto the different direction of the body relative togravity, when the rider stays on the inside of

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Figure 11. The photograph shows the Kanonen train onthe way back through the loop into the heartline roll,where the centre of mass of the rider moves essentiallyalong a straight line.

the screw and is upside down at the top. Thisobviously does not apply in situations, such asfigure 12, where the train has twisted around to thetop of the track in the highest point. Correctionsmay also arise due to the motion of the train aroundthe track. Any difference in radius of curvaturebetween the high and low points also leads to achange in g-force difference to what is expectedfrom these formulae.

Speed Monster corkscrew

The corkscrew in the Speed Monster is quitestretched, making good use of the available space,as seen from the panorama picture in figure 2. Inthe Kanonen ride, the corkscrew is stretched to thepoint where the riders move along a straight linewhile the track twists around them, giving a ratioR/L close to zero.

As an exercise, estimate R/L for the SpeedMonster corkscrew from figure 2. Use thisratio to estimate the difference in g-force for thedifferent parts of the ride. Does your result agreewith the accelerometer data in figure 5, wherethe corkscrew spans the period of about 10 s,starting at t = 28 s? Are there any deviationsfrom expectations that would prompt you to makeadditional observations or measurements in thepark?

The heartline roll of Kanonen

On the way back to the station, the Kanonen trainperforms a show-off passage over guests in thequeue (figure 11). The track turns about 270◦ in a

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0

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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.

29t(s)

333130282625–2

Kanonen, Heartline Roll

total

lateral vertical

a/g

‘heartline roll’. The body’s centre of mass moveswith nearly constant velocity. What forces act onthe body? Figure 12 shows the accelerometer datafor this part of the tour.

It is tempting to believe that measurementwith a three-dimensional accelerometer gives acomplete description of the motion, which can beused to recreate the shape of the track. However,the accelerometer data between 27 and 30 s infigure 12 could be obtained without rotation bymoving up–down and left–right, some 20 m ineach direction (although the altitude profile does,indeed, show that this was not the case).

Newton’s first law tells us that a body remainsin uniform rectilinear motion unless acted on byunbalanced forces. However, when the ‘body’in Newton’s laws is our own it is clear thatthe direction of the forces relative to the bodymatters: we are not point-like particles. A ‘motiontracker’ needs also to measure rotation aroundthe three axes to get a complete description ofthe motion [9]. Nevertheless, three-dimensionalaccelerometer data provide much material foranalysing familiar motions.

Personal experiencesThe wireless sensor system

The WDSS system is extremely simple to use.Once set up from a computer, it can be used bya large number of students to collect data overa whole afternoon. It is, however, important tokeep notes of what rides have been studied, sinceno additional information is stored on the sensor.

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Should the memory fill up, the data are quicklytransferred to a laptop, and the sensor is againready for additional measurements.

The measuring vest, although not particularlyaesthetically pleasing, seems to convince rideattendants that the wearer is serious. Mostimportantly, the vest keeps the sensor from fallingout during the ride: safety concerns must alwayscome first. A disadvantage is, however, that it isdifficult to keep the coordinate axes aligned [10]1,but in most cases this can be dealt with afterwards.

One-dimensional accelerometer data are suf-ficient to obtain velocity and position rectilinearmotion, at least in principle. For a complete de-scription of three-dimensional motion, accelerom-eter data for the three axes must be complementedby rotational data around all axes, as discussed byPendrill and Rodjegard [9], in connection with theanalysis of motion tracker data for a roller coaster.Still, the simplicity of use for the WDSS sensormakes it a useful tool, bringing the amusementpark experience to the classroom.

The roller coasters

Although a three-dimensional accelerometer recordsthe time series of forces acting on the body, it canobviously not capture the whole experience.

Part of the experience is the build-up ofexpectations during the time in the queue. TheStealth queue at Thorpe offers TV screens withits own disc jockey. During my one-hour wait toget on (August 2006), people in the queue weredancing to the music, and in general having agood time. There was also the occasional speakermessage telling NN to get ‘back to the entrancewhere mum has got a FastTrack ticket for you’.Just before entering, the riders are brought in closeview of parts of the launch technology. The queuealso lets you look up towards the 62 m high tophat (figure 3), and I have to confess that it wasthe first time in many years that I had the feeling‘am I really going to go on that ride?’. But, yes,I did, and even got a FastTrack ticket for a secondride. The long period of acceleration followed byweightlessness is quite a strong experience.

Both the Kanonen and the Speed Monsteroffer good views of different parts of the ride asthe queue moves on. The best queuing experiences

1 Figure 3 of [10] shows the spillover from the verticalaccelerometer reading, which results when the coordinate axesare not correctly aligned.

are, of course, during off-season, when you do nothave to wait more than a few minutes to board thetrain.

Rita—Queen of Speed, Kanonen and theSpeed Monster all have slower speeds and lowerhills than the Stealth. Rita—Queen of Speedreaches a higher speed (98 km h−1) than theScandinavian launch coasters, but the ride heightsat Alton Towers are limited by the tree tops. Thespeed gained from the launch is instead used in ahelix with an extended period of relatively strongg-forces. The whole 640 m tour in Rita the Ridelasts 25 s, which may seem a bit short after along time in the queue. Alton Towers has morerewarding roller coaster rides!

The Kanonen and the Speed Monster bothturn the rider upside down a few times duringthe ride, in loops and screws, discussed above.Although the inversions could be captured by arotational sensor, the visual experience could not.The Kanonen launch goes across a small river,giving the riders the impression of falling intothe water after the top hat. The Speed Monsterhas a most spectacular track layout, encircling theentrance escalators. It runs on a hillside, andbrings the rider through the terrain, close to thetree tops. The Kanonen track is woven back andforth, making maximum use of a small availablearea. Its complicated structure is more difficult tomemorize, which possibly brings more surprisesto the rider. The Speed Monster makes use of thenatural drops to bring the train considerably belowthe starting point, thereby increasing the maximumspeed. The ride is only about 2 s longer thanthe Kanonen ride, as seen from the accelerometerdata. However, the difference feels larger, possiblybecause the Speed Monster track is more than 50%longer: 690 m compared to 440 m. (The Stealthtour is even shorter: 400 m.) The longer trackalso accounts for the smoother ride, where moredistance is allowed for the different elements [8].

Which coaster is the ‘best’? To some extentthis depends on your personal preferences. Theresults from annual voting by riders can be foundat BestRollerCoasterPoll.com.

Lessons in the amusement park or roller coastersin the classroom?

Is it best to have physics lessons in the amusementpark or to have lessons in school about physicsin amusement rides? Even without easy access

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to an amusement park, most students are likelyto have been on the rides and can relate theexperience of their body to the physics descriptionof the rides. Swings in a nearby playgroundare a good way to introduce amusement parkphysics [11]. Discussions of forces in the ridesare likely to change students’ ways of thinkingduring future park visits, as many students havereported. As with all field trips [12], the learningoutcome from an amusement park visit dependsto a large extent on the preparation. SeveralInternet sites provide material for preparingamusement park visits (e.g. [13, 14]). Someparks, including Alton Towers and Thorpe Park,offer educational programmes for visiting schoolclasses. Thorpe Park also quotes educationsecretary Alan Johnson: ‘Learning outside theclassroom should be at the heart of schools’curriculums and ethos.’

Measurements in the park can easily over-shadow analysis, which is left for later. Backin the classroom, the rides are no longer at handfor investigating questions arising from the data.The balance between measurement and analysis isworth careful consideration. The analysis of mea-surement data also takes somewhat different formsdepending on what data can be obtained from thepark. Drawings are usually secret, as required bythe agreements between parks and designers. Thelength of a roller coaster train can, however, usu-ally be obtained. (If not, it can be estimated bymeasuring the width of the gates in the boardingqueues.) It can provide a length scale for analy-sis of different elements of the roller coaster fromphotographs or video clips. The length, combinedwith the time of passage at a given point, gives aspeed measurement. Comparing timing from stopwatches of a number of students’ mobile phonesprovides good material for discussions of measure-ment uncertainty. Sometimes the track layout alsomakes it possible to estimate energy losses frommeasurement of the time of passage.

I find that every time I get new data from aride, they give rise to questions, and an urge to goback and check. Now, I would like to go back toTusenfryd and take a good look at the corkscrewto see if what looks like an extra large radius ofcurvature at the bottom of one of the coils canexplain 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

the top of the first coil (figure 6, at about 28 s inthe data shown in figure 5). Measurements are notonly about numbers, but about questions, answersand insight.

Acknowledgments

First, I would like to express my appreciationto roller coaster designer Werner Stengel forkindly sharing part of his knowledge about variousaspects of roller coasters, including loop shapes.I would also like to thank Jochen Peschel fromCoasters and More for permission to use thephotograph in figure 2, and for interesting e-mailcorrespondence. Finally, I would like to thankthe helpful people at Liseberg and Tusenfryd, inparticular Ulf Johansson and Morten Bjerke, forpractical help and for stimulating discussions.

Received 2 January 2008, in final form 6 February 2008doi:10.1088/0031-9120/43/5/003

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Felshang Coasters and More www.coastersandmore.de/rides/speedmonster/speedmonster.shtml

[2] Marden D 2003 3, 2, 1, launch! Fun World Mag.www.iaapa.org/industry/funworld/2003/Jul03/Features/3 2 1 Launch!/3 2 1 Launch!.html

[3] Peschel J 2007 Xcelerator—Intamin’s acceleratorcoaster premiere Coasters and More www.coastersandmore.de/rides/xcel/xcelerator.shtml, see also www.coastersandmore.de/rides/kanonen/kanonen eng.shtml

[4] Schwarzkopf A 1979 Amusement ride withvertical track loop US Patent Specification4165695 (DE2703833)

[5] Higgins A 2003 The coaster with the mosterMachine Design www.machinedesign.com/ASP/strArticleID/55720/strSite/MDSite/viewSelectedArticle.asp

[6] Schutzmannsky K 2001 Roller Coaster—DerAchterbahn-Designer Werner StengelIngenieurburo Stengel www.rcstengel.com

[7] Pendrill A-M 2005 Roller coaster loop shapesPhys. Educ. 40 517–21

[8] Stengel W 2007 Private communication[9] Pendrill A-M and Rodjegard H 2005 A roller

coaster viewed through motion tracker dataPhys. Educ. 40 522–6

[10] Butlin C A 2006 Flying high with sensor systemPhys. Educ. 41 577–9

[11] Pendrill A-M and Williams G 2005 Swings andslides Phys. Educ. 40 527–33

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[12] Rennie L R and McClafferty T P 1996 Sciencecenters and science learning Stud. Sci. Educ.27 53

[13] Bakken C 2007 Physics/Science/MathDays@Great America http://physicsday.org

[14] Pendrill A-M 2007 Science in the Lisebergamusement park http://physics.gu.se/LISEBERG/

Ann-Marie Pendrill is professor inphysics at Goteborg University, with abackground in computational atomicphysics. Her teaching involvesengineering, physics and teacherprogrammes and she is involved withdifferent forms of informal learning,including amusement park physics.

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