Acceleration in 1, 2, and 3 dimensions in launched roller coasters

Ann-Marie Pendrill, Department of Physics, University of Göteborg Göeborg, Sweden
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 paper 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.
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])
Figure 3: The 62 m high "Top Hat" of the Stealth coaster at Thorpe Park.

1. Introduction

3, 2, 1 ... launch! The traditional lift hill, which gives the initital potential energy for the ride, is absent in some newly build 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 characterised 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 [1-4].

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 disucssed in more detail below.

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

Figure 6: The launch of the Speed Monster, with a side view of the "Norwegian loop".

2. One-Dimensional Horizontal Motion

The study of motion in school 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 illustration to textbook presentations. In one dimension, the measurement of the acceleration 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 peformed numerically or analytically, after approximation of the acceleration time dependence. The hydraulic launch also gives an example of applications of the gas laws, as discussed in Section 2.2.

2.1 The launch

Flags in the launch area enhance the sensation of motion during launch of the Speed Monster, as shown in Figure 6. Horizontal launch of roller coasters have been used since the 70s, e.g. in the Revolution [2,3] which is a Schwarzkopf "shuttle launch coaster" [5], where the energy is stored in a flywheel. Magnetic launch techniques were introduced during the 90s, with "LIM" - Linear Induction Motors" and "LSM" - Linear Synchronous motors. Compressed air launch was introduced in 2002, followed by 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. [2,3,6].

In the hydraulic launch, oil is pumped from a reservoir into storage cylinders filled with nitrogen. The energy is built up as the 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 [4], who also presents an animation of the launch process.

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 4 cars weighs about 8 tonnes. The Speed Monster train with 3 cars is lighter, about 6 tonnes. These weights include the mass of the sled used during acceleration.

Exercises for the reader:

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

2.2. Gas Pressure During the Launch

The launch, itself, can be assumed to be an adiabatic process, where
For a diatomic gas, like nitrogen, used for the launch, =1.4. The work done by the gas during adiabatic expansion is given by

According to the gas law, pV/T=constant, and this expression can be used to relate the temperature during the adiabatic expansion to the pressure or the volume.

The relations above can be expressed in terms of dimensionless quantities depending on the ratio V/Vo. The maximum possible work for an adiabatically expanding gas would be Wo=poVo/(gamma -1), which depends on the initial pressure and volume. The drop in acceleration in Figures 4 and 5 correspond 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.
Figure 9: Photo showing the Speed Monster train just before it enters the brakes. Braking is accomplished by magnets on the train inducing eddy currents in the metal sheets sticking up in the last parts of the track. Figure 10: Acceleration, velocity and distance during the braking of the Speed Monster train. The negative accelerometer data indicate a retardation of the train.

2.3 Stopping the train

The last section of the Speed Monster track is shown in Figure 9, which also shows the magnetic brakes in the last part of the track. Eddy currents are induced by rare-earth magnets on the train in the conducting metal sheets, the "braking swords". Magnetic brakes are also mounted on the launch track (see Figure 6). These are taken down just before launch, but automatically rise when the train has passed, to prevent the train from crashing back into the station, should the initial energy be insufficient to reach the point of no return at top of the first hill. [6].

Magnetic brakes offer at smooth onset of the braking, as a successively larger fraction of the magnets on the train come close to the braking swords. Since the magnetic field induced in the swords is proportional to the velocity of the magnets, and thus the train, the braking force is then reduced with time after all parts of the train have entered the braking region, leading to an exponentially decaying speed. The kinetic energy of the train is converted to heat in the metal sheets which cool off again well before the next train arrives. This will be analysed in more detail in a separate paper.

Both in Kanonen and in the Stealth, magnetic brakes are also mounted at the end of the camel back which follows the initial "Top Hat". The Speed Monster train, on the other hand, runs the whole length of the track before encountering the final brakes. The accelerometer data from the braking are shown in Figure 10, together with graphs for velocity and distance, obtained from the data by assuming that the train has come to a stop at the end. Since the longitudinal acceleration component is measured along the direction of motion, a negative value indicates a retardation of the train. By integrating the accelerometer data for the braking of the Speed Monster train, we find that the train has entered the final brakes with speed of 18m/s, compared to leaving the launch part with a speed of 22m/s. (What fraction of the initial kinetic energy is lost during the ride?)

2.4 Energy Losses

How much energy is lost as the train moves along the track? The layout of the Speed Monster makes it possible to obtain an estimate, by using the Speed Monster panorama photo in Figure 2 which shows how the hills and valleys of the corkscrew become slightly lower with every coil. (Technically, only the last hill is considered as part of a corkscrew since the torsion of the track in the first two hills brings the train to the top of the track without any inversion.)

Make a larger copy of the photo and draw a line beteen the highest points of the track. You can also draw a similar line between the lowest points. Measure (in mm on the photo) how much the altitude, H, changes for a given horizontal distance, D. Before working out a number, we should also account for the additional distance due to the circular motion. The diameter, 2R, of the circle can be obtained from the distance between the lines connecting the upper and lower points, respectively. As the train moves a distance L to the right, while completing a whole turn, it also moves a distance 2R around the circle, giving a total distance


The ratio
can be taken as an estimate of the "effective" friction coefficient, which includes air resistance and other losses. For cold and empty trains, the relative energy losses can be more larger, as much as 4cm/m [10]. This must, of course, be allowed for in roller coaster design, so that even these trains can make it safely back to the station.

Exercise: How does the estimate of the energy losses from the photo agree with the comparison of train speed just after launch and just before the brakes? (At this point the track lies 1m lower than the track directly after launch [10], and the track length is 690m.)

3. Acceleration measurements in three dimensions

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 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 "g-force".

The accelerometer data in this paper were obtained using a wireless dynamic sensor system [7]. This system also also measures 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, e.g. around launch in the graphs in Figures 4 and 5.)

3.1 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 this one, the x-axis concides with the direction of motion. The y-axis gives the direction of "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 elements in a roller coaster, lateral components vanish if the curves are perfectly banked.

A problem in measuring acceleration in three dimension 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 shows a comparison of the total g-force and the vertical component.

When also the other coordinates are of interest, as for launch, break and roll, is necessary to perform a coordinate transformation. The data in this paper were transformed by rotating the axes so that the data has only a vertical component before the ride starts, and assuming that the sensor orientation relative to the track is fixed.

Figure 11: The "Norwegian Loop" of the Speed Monster coaster encircles the roller coaster entrance to the park, making possible a very large loop. In view of the short Speed Monster train, the ratio between train length and loop radius thus becomes unusually small in this case.

4. Two-Dimensional Motion in Loops

Both the Kanonen and the Speed Monster include loops, where the train moves essentially in two dimensions. The photos in Figure 1 and 11 show that neither the loop in Kanonen nor in Tusenfryd 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 as discussed in more detail in [8,9]. The Kanonen loop is a classic "clothoid loop", which was introduced by Werner Stengel in 1976 in the roller coaster Revolution [2,8].

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 [2]. It gives the rider two inversions, both during entrance and exit from the loop.

The Kanonen train passes the highest point at the 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 10s and 15s, 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.

Figure 12: The large corkscrew of the Speed Monster. Technically, only the last hill is considered as a corkscrew element, as noted in Section 2.4: 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=33s in the graph in Figure 5.

5. 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 Figure 2 and Figure 12.

A corkscrew can, as a first approximation, be described in cylindrical coordinates, where the circular motion with a radius R is then accompanied by a perpendicular motion along the cylinder axis. In the photo of the corkscrew in Figure 12 the track seems a bit flattened at the top. At the same time, the track twists, so that 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 2 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

Exercise: 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 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 formulæ.

5.1 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 12. 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 seconds, starting at t=28s. Are there any deviations from expectations that would prompt you to additional observations or measurements in the park?

Figur 13: The photo shows the Kanonen train on the way back through the loop into the heartline roll, where the center of mass of the rider moves essentially along a straight line.

5.2 The Heartline Roll of Kanonen

During the way back to the station, the Kanonen train performs a show-off passage over guests in the queue (Figure 13). The track turns about 270 degrees in a "heartline roll". The body's centre-of-mass moves with nearly constant velocity. What forces act on the body? Figure 14 shows the accelerometer data for this part of the tour.

It is tempting to beliew 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 14 could be obtained without rotation by moving up-down and left-right, some twenty meters in each direction (although the altitude profile does, indeed, show that this was not the case).

Newton's first law tells us that 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 to measure also rotation around the three axes to get a complete description of the motion. [11]. Nevertheless, three-dimensional accelerometer data provide much material for analysing familiar motions.

Figur 14: Accelerometer data for the "heartline roll". The vertical and lateral components are shown together with the total g-force 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 througout the roll. However, the direction is changed relative to the rotating coordinate system of the body and of the accelerometer.

6 Personal experiences

6.1. The Wireless Sensor System

The WDSS system [7] is extremely simple to use. Once setup from the computer, if 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. Should the memory fill up, the data are quickly transfered to a laptop, and the sensor is again ready for additional measurements.

The measuring vest, although not particularly æsthetically 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 [12], 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. In three dimensions, accelerometer data for three axes must be complemented by rotational data around all axes, for a complete description of the motion, as discussed by Pendrill and Rödjegård [11], 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.

6.2 The Coasters

Although a 3D accelerometer records the time series of forces acting on the body, it can obviously not capture the whole experience.

Part of the experience is the build-up of expectations during the time in the queue. The Stealth queue at Thorpe offers TV screens with its own disc-jockey. During my one-hour wait to get on (Aug 2006), people in the queue were dancing to the music, in general having a good time. There was also the occasional speaker message telling NN to get "back to the entrance where mum has got a FastTrack ticket for you". Just before entering, the riders are brought in close view of parts of the launch technology. The queue also lets you look up toward the 62 m high Top Hat (Figure 3), and I have to confess that it was the 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 second ride. The long period of acceleration followed by weightlessness is quite a strong experience.

Both the Kanonen and the Speed Monster offer good views of different parts of the ride as the queue moves on. The best queueing experiences are, of course, during off-season, when you don't 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) 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 seconds, 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 entrance escalators. It runs on a hillside, and brings the rider through the terrain, close to the tree tops. The Kanonen track is woven back and forth, making maximum use of a small available area. Its complicated structure is more difficult to memorise, which possibly brings more surprises to the rider. The Speed Monster makes use of the natural drops to bring the train considerably below the starting point, thereby increasing the maximum speed. The ride is only about 2 s longer than the Kanonen ride, as seen from the accelerometer data. However, the difference feels larger, possibly because the Speed Monster track is more than 50% longer: 690 m compared to 440m. (The Stealth tour is even shorter, 400 m.) The longer track also accounts for the smoother ride, where more distance is allowed for the different elements [10].

Which coaster is the "best"? To some extent this depends on your personal preferences. The results from annual voting by riders can be found at BestCoasterPoll [13].

6.3 Lessons in the Amusement Park or Roller Coasters into the Class Room?

Is it best to have physics lessons in the amusement park or to have lessons in school about physics in amusement rides? Even without easy access 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 [14]. Disucssion of forces in the rides are likely to change students' way of thinking during future park visits, as many students have told. As with all field trips [15], the learning outcome from an amusement park visit depends to a large extent on the preparation. Several www sites provide material for preparing amusement park visits [e.g. 16, 17]. Some parks, including Alton Towers and Thorpe Park, offer educational programs for visiting school classes [18,19]. Thorpe Park also quotes education secretary Alan Johnson "Learning outside the classroom should be a 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, on demand from the park, the designer, or both. 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 photos 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. I would like to time the train as it comes around the last curve, and possibly also make a video clip, to check the speed estimate in Figure 10. I would also 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 10 - 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 in the data shown in Figure 5). Measurements are not only about numbers, but about questions, answers and insight.


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, trim brakes and energy losses. I would also like to thank Jochen Peschel from Coasters and More for the permission to use the photo 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.


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AMP, draft, 10 August 2007, updated 28 Dec 2007