# Roller Coaster Physics Coursework Ideas

An amusement park or a fun fair is an ideal place for physics experiments that could never be performed in the classroom. Several projects worldwide have developed suitable teaching activities (see ‘Resources’).

One of them is ‘*Mirabilandia, un’aula senza pareti*’ (‘Mirabilandia, a classroom without walls’), an Italian project started in 2002 by teachers from the secondary school Liceo Torricelli in Faenza, at their local amusement park, Mirabilandia^{w1}. The initiative was so successful that the park’s management decided to make it available to all schools, providing educational tours by trained science students.

Maybe you could discuss a similar arrangement with your local amusement park? Meanwhile, here are some activities suitable for any amusement park or fun fair. For some, you will first need to find out about rides’ technical specifications, either from the park or online, for example at the Roller Coaster Database^{w2}. To take measuring instruments on board a ride, safely tied to the students’ wrists or the ride’s structure, you will need the park authorities’ consent. In our case, the school’s headmaster contacted the park management, detailing the project we had in mind.

The activities are suitable for ages 15-19, but can be adapted for younger students. They cover the topics of rectilinear and circular motion, speed and acceleration, the three laws of motion, gravity, centripetal and centrifugal forces, and energy transformation.

Students should be familiar with the basic concepts of kinematics and dynamics, and will need a basic introduction to the activities before they visit the park.

Work on each ride takes about one hour, but queuing for the ride may increase the time required. One team can perform the measurements on board, while others work on the ground. Usually there is no time for each group to repeat measurements, but data from different groups can be compared. Preliminary evaluations can be made at the park and continued back in class.

At Mirabilandia, students receive a workbook to fill in. Part of this has been translated into English and is available for download (see additional material on the right).

## Roller coasters

### On the ground

Draw a rough two-dimensional height profile of the roller coaster and discuss the energy transformations along the way, especially on the first descent and vertical loops.

At the top of the roller coaster, before the first descent, the gravitational potential energy is at its maximum; along the descent, which often has a parabolic trajectory where passengers can experience weightlessness for a few seconds (in an ideal case without friction, the vertical component of the motion increases with the constant acceleration *g* due to gravity as in free fall), this is transformed into kinetic energy.

Use vertical loops to discuss the role of gravity versus centrifugal force. The loops are drop-shaped, following the course of a *clothoid *curve, in which the radius of curvature is in inverse proportion to the distance from the centre, thereby lowering the centripetal accelerations prevailing in the loop to make it more comfortable for the passengers.

You can only really appreciate how fast a roller coaster moves by determining its speed:

a) Calculate the instantaneous speed by dividing the length of the roller coaster train by the time it takes for the whole train to pass a specific point along the ride.

b) To obtain the average speed of the train along a section of the roller-coaster track, choose one that is easy to see and simple in terms of shape. Count the number of sleepers (regularly spaced bars between rails) on that section and subtract one to get the number of intervals. Time how long it takes for the train to travel along this section of track. To calculate the distance travelled, multiply the number of intervals by the interval length (the distance between two sleepers). Divide this by the time taken.

Compare the instantaneous speed recorded at the end of the first (fastest) descent with the speed calculated for a body in free fall: v= √(2 *g *h) where v: speed, *g*: gravity acceleration, h: height.

The measured speed should be lower, due to friction.

### On the roller coaster

Pay attention to sensations of lightness or heaviness (acceleration) you may experience along certain sections of the ride. More accurate data can be obtained with spring accelerometers, or more complex handheld instruments^{w4}. Instructions for how to build and / or obtain these and evaluate your results are available for download (see right).

Recently, smart phones including a three-axis accelerometer have become more popular. Applications for these phones allow you to draw graphs of acceleration over time or to measure tilt angles. These involve fewer safety issues since they can be kept inside a pocket.

## Drop towers

Drop towers are vertical tracks up and down which the passengers travel, seated on a gondola. In one type, the gondola is dropped in free fall; in the other, it is pushed down or up with a force greater than that of gravity.

From the ground, you can calculate the average speed of the gondola by timing its descent with a stopwatch and dividing that by the height of the tower. This can be compared to the instantaneous speed of a body in free fall. To measure the height of the tower, use a protractor and basic trigonometry or geometry. Detailed instructions are available for download^{w3}.

During the ride, you can feel the acceleration in your body and measure it with portable electronic instruments^{w4}.

On towers where the gondola is pushed down, the acceleration can reach up to three times that due to gravity.

If you take a plastic cup full of water along, you can observe that in free fall, the water will remain inside the cup, but if the gondola is thrust downwards rather than falling, the water level in the cup will rise. Alternatively, you can tie a small rubber ball to your wrist (pierce the ball to attach the string) and place it on your palm.

## Ferris wheels

The following activities are also suitable for younger students, aged 11-14. They should be familiar with the concepts of speed and atmospheric pressure, and will learn about uniform circular motion (time, speed, frequency) and apply their knowledge about atmospheric pressure. Either on the ground or during the ride, measure and note down the time after a quarter turn, half a turn, a three-quarter turn and a complete turn of the wheel.

The students should find that the time taken for each quarter of a turn is similar, therefore the motion of the wheel is regular and the speed constant.

Calculate the length of the gondola’s path (from the radius of the wheel), its average speed, and, as a maths exercise applied to the real world, the angle at the centre of the wheel between two bars that connect two neighbouring gondolas to the centre (divide 360° by the number of gondolas).

You can determine the wheel’s height in the same way as that of the drop tower (see above) or by using electronic barometers to determine the change in atmospheric pressure during the ride.

Every 0.1 kPa of pressure change corresponds to about 8 m of altitude change. A more accurate description of the relationship between pressure and altitude is available online^{w5}. Compare the results and discuss the advantages and disadvantages of each method (accuracy, instruments, time and effort involved).

## Carousels

A carousel is the perfect setting to study Foucault’s pendulum and the Coriolis effect. Students should be familiar with the motion of a pendulum, the laws of motion, gravity, centripetal / centrifugal forces, and rotation. Details of how to perform these experiments are available for download (see right).

## Web references

- w1 – To learn more about Mirabilandia, see: www.mirabilandia.it
In early September each year, Mirabilandia introduces their educational tours during an open day for teachers. See: www.mirabilandia.it/#/scuole

- w2 – To access the Roller Coaster Database, see: www.rcdb.com
- w3 – For instructions for measuring heights using trigonometry and geometry; studying Foucault’s pendulum and the Coriolis effect with the help of a carousel; and details of how to build your own spring accelerometer or data collection kit, see the downloadable file: PDF or Word
^{®}formatFor an English translation of part of the Mirabilandia student workbook, see the downloadable file: PDF or Word

^{®}format.

- w4 – The US companies Vernier and Pasco offer dedicated measuring instruments for use in amusement parks, which come with a full set of instructions and activities. See:
- w5 – For the relationship between atmospheric pressure and altitude, see the website of the atmospheric chemistry department of the Max Planck Institute for Chemistry in Mainz, Germany (www.atmosphere.mpg.de) or use the direct link: http://tinyurl.com/pressure-altitude

## Resources

- For a detailed report on the project in Italian, see: http://kidslink.bo.cnr.it/irrsaeer/borsric02/fisica.zip
- Similar activities to Mirabilandia take place at the Liseberg Park in Göteborg, Sweden, led by physicist Ann-Marie Pendrill. For more information (in English and Swedish), see: http://physics.gu.se/LISEBERG
- The Parque de Atracciones Madrid in Spain offers its physics and teaching handbooks (in Spanish) for download. See: www.parquedeatracciones.es/aulas-fisica
- In the USA, physics days for schools in amusement parks, including roller-coaster design contests and worksheets, are a common concept. See:
- For an interactive guide to the physics of common amusement park attractions and an activity to design your own roller coaster, see: www.learner.org/interactives/parkphysics
- For more inspiration, see:
Alberghi S et al. (2007) Is it more thrilling to ride at the front or the back of a roller coaster?

*The Physics Teacher***45(9)**: 536-541. The article is freely available online.Bakken C (2011)

*Amusement Park Physics*. College Park, MD, USA: American Association of Physics Teachers. ISBN: 9781931024129Unterman NA (2001)

*Amusement Park Physics: A Teacher’s Guide*. Portland, ME, USA: J Weston Walch. ISBN: 9780825142642

- To find out more about roller coasters, see: http://tlc.howstuffworks.com/family/roller-coaster.htm

## Author

Before his retirement, Giovanni Pezzi taught physics in secondary schools in Italy and was a supervisor for the postgraduate school of teacher training at the University of Bologna. He has written many articles about physics teaching for education magazines, and contributed to physics and computer science textbooks. He holds workshops and lectures on updating teaching methods and designed and developed maths and physics activities for the launch of the Mirabilandia project.

CC-BY-NC-SA

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When amusement rides are studied in the physics classroom, the influence of air is most often neglected. However, in the new sky roller ride [1], the air plays a crucial role for inducing the rolling motion. The wings can be rotated so that the front of the wing points up or down. With different orientation of the two wings, the motion through the air leads to a torque around the direction of motion, as the ring of planes moves around the tower (figure 1). The direction of the torque is reversed when the orientations of the wings are changed. With practice, it is possible to change the orientation in phase with the motion, so that the plane makes larger and larger oscillations until it makes a full turn around its axis. It can then continue to spin ('roll') if the wings are held in position.

This motion can be studied with, e.g. with a smartphone [2–4], which can measure rotation, as well as acceleration, in a coordinate system moving and rotating together with the rider. Only a decade ago, rotation measurements were a specialised task [5], well beyond the capabilities of most school laboratories, and they are still rarely used in education. In the sky roller ride, rotations are in focus, and this paper shows examples of measurements and interpretations of the data. Figure 2 shows one of the planes seen from front, together with a definition of the coordinate axes used to describe the motion in a comoving system. Rotations around the *x*, *y* and *z* axes are referred to as roll, pitch and yaw, respectively.

### 2.1. Around the tower

Figure 1 shows a SkyRoller ride in motion. A plane in the outer ring moves in a circle with a radius m. This value is obtained from the total diameter, 21.2 m [1] but accounting for the horizontal distance between the center of the plane and the outer wing tip. According to the data sheet [1], the plane moves around the tower 11 times min^{−1}, giving a period of s. We can also introduce an angular velocity , where the negative sign holds for the clockwise motion and the stationary axis *e*_{Z} marks the upwards direction of the tower. We thus find a speed m s^{−1} for the planes in the outer ring. The acceleration due to this circular motion is given by . A park can also choose to run the ride with 9.5 or 8.5 turns min^{−1} [6], which would lead to correspondingly smaller accelerations. Smaller values for velocity and acceleration are also obtained for planes in the inner ring.

To counteract the force, , due to gravity, while also providing the force necessary for the acceleration of a rider with mass *m*, the force from the ride must be . For a centripetal acceleration , the size of the force from the ride becomes 1.6 mg, directed upwards and towards the centre, at an angle of about from the vertical, as in figure 3. A rider will perceive the direction of this force from the ride as 'up', (figure 3), due to the equivalence principle, as discussed, e.g. in [7–9].

### 2.2. Sensor data, acceleration, rotation and coordinate axes

Figure 4 shows data collected with smartphone built-in sensors [3] during the start of the ride in a one of the planes closest to the tower, with an estimated radius of rotation m. It gives a first illustration of the forces and rotations, expressed in the comoving coordinates. The air pressure gives an indication of elevation, and drops from 1016.7 hPa to 1013.2 hPa as the ride moves toward the top of the tower. This change in air pressure can be converted to an elevation gain, giving m.

As long as the planes (and sensors) do not tilt, the rotation is only around the vertical axis, both in the stationary and rotating system. This rotation in relation to the body is referred to as 'yaw', and the negative sign indicates clockwise motion. From the graphs we can see that the ride starts moving upwards before the rotation starts, at around 93 s in the graph. As the plane starts to move faster around the tower the total force on the rider becomes larger, as seen by the increasing red curve in the top figure. However, the sideways (lateral) force remains close to zero—since the plane is free to rotate, to align with the vector . The tilt of the plane is most evident in the angular velocity curve for the 'pitch', which rises around 18 s, before the effect is clearly noticeable in the force on the rider. The rotation axis, *e*_{Z}, for the main rotation around the tower can be expressed in terms of the comoving coordinates as , if the plane is tilted by an angle *θ*. Since the angular velocity around the tower is negative and the angle *θ* is positive, both the *z* component (yaw) and the *y* component (pitch) are negative.

### 2.3. Oscillations around the equilibrium angle

To start the rolling of the plane, the wings must be tilted as in figure 5. The torque from the oncoming air for a tilt by an angle is proportional to , which has a maximum [11] for , which is larger than the maximum tilt angles allowed by the construction ( for airplanes in the outer circle and for the inner circle [6]).

By keeping the two wings in opposite orientation, a torque *τ* can be exerted on the plane, causing it to rotate slightly from the equilibrium angle defined by direction of the the vector . As soon as the plane is at an angle *θ* from this equilibrium orientation, there is an opposing torque, due to gravity and acceleration. This opposing torque causes the rotation to slow down and stop and then, the plane to swing back, past the equilibrium orientation. When the plane changes its direction of rotation, the orientations of the wings must be reversed, and the plane can then rotate to a larger angle next time. With experience and practice, this change can be made more quickly and timely, leading to faster increase in the maximum angle of the swing, and ultimately to a fast rolling.

Figure 6 shows the angular velocity together with the 'vertical' and 'lateral' components of the vector for the start of a ride, up to the transition to fast rolling. These data were collected by one of the authors (ML) with a home-built dedicated system built into a small custom-made case (figure 7).

The total torque on the airplane can be expressed as , where *M* is the mass of the plane (including the rider), *h* is the distance between the center of mass and the axis of rotation, and *τ* is the torque from the two wings which is independent of the angle , but changes sign when the orientations of the wings are interchanged. It should also be noted that the torque from the wings is reduced for fast rotations, since the rotations reduce the relative velocity between air and wings.

The equation for the time dependence of the angle becomes

The dots denote time derivatives, but we can also use also the notation for the angular velocity and for the angular acceleration. This formula can be used to obtain an expression for the rotational energy , using the relation

analogous to the expression for linear motion. We show in the appendix how this equation can be used for a more detailed investigation of the initial rolls.

### 2.4. Data for the full ride

Figure 8 shows the lateral component of the accelerometer data for the full ride, together with data from the first ride of one of the authors (AMP), collected with a wireless dynamic sensor system (WDSS) carried in a data vest [10]. The lateral component was chosen for the graphs in figure 8, since it is more sensitive to small changes in angle. Smartphone sensors (see e.g. [2–4]), which typically measure up to along each axis, are also well suited for this ride, with its relatively small forces.

Figure 8 provides a comparison between a ride where the planes never makes it all the way round and one ride with more than 40 full rolls. Fast rolling requires practice in order to achieve the right timing and the fast reversal of the orientation of the wings.

A first observation is that the maximum lateral force is about 1.6 *mg*, both for the limited turns and for the full turns, where the angular velocity is much larger. Also the vertical components reach essentially the same value (positive, as well as negative). The centripetal acceleration connected with the rolling of the plane must thus be negligibly small. This reflects the position of the sensors close to the axis of rotation. The value for the lateral component thus depends only on the angle relative to the force from the ride.

The angular velocity around the *x* axis is of particular interest in the SkyRoller ride. This is also accessible through smartphone sensors [4]. Figure 9 shows the angular velocity data from the rolling ride together with the accelerometer data from the lower graph in figure 8, both obtained using the home-built system shown in figure 7.

The main rolling motion happens with a period around 1 s, corresponding to an angular velocity of magnitude around , consistent with the results in the graph. In addition, we can note a small variation in the amplitude, with a period around 6 s. A headwind causes a larger torque, and a slightly faster rotation, so this period reflects the time for a full turn around the tower.

Figure 10 shows in more detail a sequence of the fast rolling. The angular velocity, *ω*, varies during a full turn, and the resulting variation in rotational energy reflects the change in potential energy between the 'upright' and 'upside-down' positions. The difference between maximum and minimum angular velocity during the spinning also gives an indication of the critical angular velocity required at the bottom to make a full turn. Choosing e.g. the oscillations around the time 66 s in figure 10 we obtain an estimate

This can be compared to the angular velocity just before the time 30 s in figure 9.

We note that the slope of the accelerometer data is slightly smaller when going from negative values (right side up) to positive values (left side up) than vice versa. The smaller slope corresponds to a slower rotation, i.e. with the plane 'upside down' and this rotation pattern is consistent with a negative angular velocity around the *x* axis, as seen also in figure 11, which shows the rotation of coordinate axes in more detail for one period. A positive sign corresponds to the motion of a right-handed screw. The motion illustrated in figure 11 is instead in the negative *x* direction, consistent with the negative sign of the angular velocity for rolling. We can also note that the angular velocity keeps getting more negative for a short while after the lateral force is zero as the plane passes the 'lowest' point. This is due to the torque of the wings that continues to add energy.

### 3.1. Limits to the angular velocity

During the fast rotations, the wings are kept in the same position, and we might have expected a continued increase of angular velocity, if the torque had remained constant. However, for fast rotations, the speed of the outer parts of the wings becomes comparable to the speed of the plane relative to the air, leading to a reduction of the torque. (The wing tip is more than *r* = 1.3 m away from the *x* axis. The speed of the wing tip due to the rotation becomes, m s^{−1} for the fastest rotation.) As the force from the air is proportional to the square of the speed relative to the surface, this can have a large effect on the rolling, leading in effect to a limitation of the angular velocity. The approximations used in the attempt to model the motion are thus valid only for small angular velocities.

### 3.2. Acceleration of the head during the rolling

The centripetal acceleration due to the rotation around the *x* axis does not have a significant influence on the forces measured with a sensor close to the rotation axis. However, the fast spinning causes a sufficiently large acceleration of the centre of mass to keep the body pushed into the seat also in the upside down position (or, rather, the seat pushes the body to provide the additional force required for the centre-of-mass acceleration). In fact, the rotation around an axis close to the heartline can be seen as a body centrifuge. The acceleration of the head, can be estimated using the measured angular velocities:

towards the center of rotation at a distance *H*. For most of the turns, the angular velocity at the 'bottom', when the plane and rider are 'upright' is around s^{−1}, which corresponds to an acceleration *g* for a distance m. The force on the head required for this acceleration needs to be added—as a vector—to the force required for the accelerated motion of the heartline, which is about 1.6 mg pointing up and towards the centre as shown in figure 3. The body then needs to pull the head (with mass *m*_{H}) towards the body with a force . This corresponds to a 'negative *g* force', −0.2*g*. For the faster rotations, the angular velocity reaches s^{−1} in the 'upright' position, and s^{−1} in the upside-down position. The top of the head then experiences a negative *g* force around −3.4*g*.

Safety standards for amusement rides typically limit negative vertical *g* forces to −2*g*. However, in a rotation around the heartline the lower part of the body accelerates in the opposite direction. We are not aware of standards that apply for rotations around the heartline. However, we may consider the rotation-induced pressure in a liquid with density *ρ* at a distance *r* from the axis, rotating with an angular velocity *ω* can be written as . This be compared with the pressure of a stationary liquid at a level *h* below the surface. Even for the largest angular velocities shown in figure 9, the rotation-induced pressure at 0.5 m from the axis is equivalent to the pressure at 0.8 m below the surface. This is thus significantly less than if the whole body is upside down or exposed to −1*g* in other ways. Still, some riders report that they experience a light headache after having achieved a large number of spins. The effect from the rotation around the heartline is added to the forces on a point on the rotation axis.

A ride, such as the sky roller invites discussion of many aspects of physics. The forces during motion without rolling are analogous to forces in a chain flyer ride [8]. The rotations of the plane bring in angular velocities, torque and rotational energies as interesting concepts. The possibility to use the wings to control the motion adds many additional challenges, e.g. identifying the direction of the force from the air on the wings, and how the force, as well as the resulting torque, depends on the tilting angle of the wing.

Modern technology offers many opportunities to collect data during the ride. A combination of accelerometer and gyroscopic data provides possibilities to check consistency of the data. However, for an understanding of data collected, a thorough discussion of the directions and coordinate axes is essential. In the sky roller ride, the motion of the body to change the orientation of the wings may also change the orientation of the axes of a sensor carried on the upper part of the body, bringing in additional complications. The analysis can be performed at many different levels of difficulty, with different types of mathematics and physics in focus, and many different types of educational assignments and student problems.

We would like to thank Kenneth Berndtsson at Liseberg for providing additional information about the ride. We would also like to thank Liseberg for invitations to pre-opening events for the Aerospin ride, when we collected most of the data, and for permission to use screen shots from the instruction movie [12]. Discussions with Werner Stengel and David Eager about amusement ride safety standards, and with Michael Axelsson about blood pressure in amusement rides, are gratefully acknowledged.

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