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, )|
|Figure 3: The 62 m high "Top Hat" of the Stealth coaster at Thorpe Park.|
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".|
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 , 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).|
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
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.|
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
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
final brakes with speed of 18m/s, compared to leaving the launch part
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
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
.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 . 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 , and the track length is 690m.)
The accelerometer data in this paper were obtained using a wireless dynamic sensor system . 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.)
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.|
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 . 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.|
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
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.|
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. . 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.|
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 , 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 , 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.
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 .
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 .
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.