Amusement Park Physics/Precalculus

Assessment

Assessment for the Amusement Park Physics/Precalculus project are in the following areas:

• data collection notebook from the amusement park.

• presentation of data from one amusement park attraction.

• merging of multiple data sets collected by all groups for one/more attractions.

• group activity/interaction.

• reaction paper

• exam.

Data Collection Notebook

Students are expected to keep a "notebook" while at the park. This notebook should contain the information about the ride, where the data was collected on the ride, and the feelings which the student experienced on the amusement park attraction. Each member of the group is responsible for his/her notebook. Our sample notebook (pdf form) contains hints and information provided by the park along with suggestions we provided for collecting data. You will note that the scrambler had some required math measurements for use when returning to class.

Evaluation consists of the students doing the required number of amusement park attractions and making detailed notes about the ride. This allows for better discussion and assimilation of the data at a later time.

Data Presentation - Groups

The first day back from the trip, we asked each group to list the rides in order for which they had the best data and could make the best presentation. Each group was assigned the ride with their top priority assuming making sure that no groups used the same attraction.

The first assignment was to prepare and present a preliminary whiteboard. The presentation is informal with a sketch of the data showing where the ride was positioned at a given time. The students were to present ideas about mathematical models which they might explore along with physics concepts which might apply to the situation. No work was required at this point. The purpose was to get ideas from other students and teachers about further areas of exploration which might be made.

Once the informal presentations were made, students returned to prepare a formal group presentation including mathematics and physics concepts. A variety of the rides are discussed below. The data sets are available for most rides but may not be the data sets used by the students in the presentations. A discussion of the analysis will also not be presented to allow other students the same opportunity to analyze and provide meaning to the data. The following is a brief description of the ride and the data collected. This is not meant to be a full investigation of the data as we choose to allow other students to experience and work through the data on their own.

Detonator

The Detonator is an attraction which propels the rider up a vertical tower. The rider then is propelled back down the tower resulting in negative g's at the top of the tower. As the rider experiences a "bungee" type ride, the acceleration data is recorded. The student sketch of the data is shown on the white board at the left during their presentation.

The data was collected on a single vertical axis with 512 data points every 0.1 seconds. The remainder of the data after the ride came to rest was purged. A 5% prestore was used to allow the trigger to be pressed after motion occurred and still get the initial data. Maximal acceleration on this ride occurs when the rider is at the bottom of the ride and minimal accelerations occur at the top of the ride. One will not that negative accelerations are the result of the ride propelling the rider downward at the peak of the first time up the tower. The second peak at approximately 12 seconds occur when the rider is back at the low point on the ride. Data in the file detonat is availabe in text file format , TI-83 Format and Vernier Graphical Analysis format. (Make sure you browser is set to save a file extension *.dat).

Scandia Scrambler

The Scandia Scrambler shown in the back has a center arm which rotates in one direction while the gondola rotates in the opposite direction. Knowing the length of the center arm (4.4 m as provided by Worlds of Fun) and the gondola arm (3.2 m), students can determine the rpm's for both the center arm and the gondola arms. Using parametric equations, the position vs. time graph can be graphed. Students can predict the time interval between maximal and minimal accelerations and compare to the recorded data. This ride uses no direct vertical motion although the students reported that the ride does move up and down and some minimal acceleration is experienced due to this motion.

The acceleration graph is shown on the graph at the right. By downloading and graphing the data on individual axes, one will see that there is a definite variation in the x (forward) and z (lateral) axes. The y (vertical) axis shows some variation but not the amplitude of the x and z axes. A second area of investigation is the position vs time graph of the Scrambler. The pattern shown below is the parametric model using the arm lengths provided by Worlds of Fun and will vary with the rpm's of the ride. Position vs. Time Model

Can you determine what points on the position vs time model (above left) relate to the acceleration vs time graph above? As one can see by the whiteboard being prepared at the left, the students are looking at a sinusoidal mathematical model along with concepts of centripetal acceleration. What does the amplitude of the acceleration graph above indicate? What is the period of the data? How does this period relate to the motion of the Scandia Scrambler? Download the data in text format, TI-83 format, or Graphical Analysis format.

Finnish Fling

The Finnish Fling (not pictured) is a ride which spins the rider in a circle and then allows the floor to drop out from beneat the rider. The graph at the right shows the data collected for approximately 60 seconds from the time the ride began spinning. Students noted that the floor dropped out from under them and they felt a vertical shift noted by the graph in purple. They believe this occurred at about 54 seconds into the data collection. If the X-axis were pointed directly toward the center of the spinning circle, what would you expect as the ride increased its angular velocity? What about a y-axis which was exactly vertical. If the z-axis is pointed in the direction of motion, what would you expect as the ride increased its angular velocity? What would happen to the z-axis as the ride reached its operating velocity? What would happen to each axis as the ride slowed?

The student presentations involved whiteboards with each member of the group discussing the findings. At times, the data was also displayed and discussed using a television monitor (upper right corner of image) connected to the computer so that group members could respond to student/teacher questions by looking at other sections of the data. Questions to ponder: If the ride has a diameter of 4.25 m and operates at 32rpm's, what is the centripetal force? What is the frictional force required to keep a rider with a mass of 80 kg from sliding down the wall? Students also found that at times it was necessary to do further investigations to determine exactly what was happening with the data and if some aspect of the data was true in other situations. In the image at the left below, the students are setting up the accelerometer on a turntable to further investigate the acceleration of circular motion. The data for this ride is available as fling in either text format, TI-83 format or Vernier Graphical Analysis format.

Omegatron

The Omegatron (not pictured) is an amusement park attraction which makes vertical circles with the rider in which the rider is upside down at the top of the circle. The graph at the ride shows the combined acceleration of all three axes. Note the sinusoidal nature of the function. The construction of the accelerometer is such that the x and y axes are perpendicular to each other. When the accelerometer is moved in a circle, the axes remain perpendicular to each other, but the y-axis is not always pointed down and the x-axis is not always pointed forward. See the next image and description for a further discussion of this.

The direction of one possible configuration for the accelerometer is show in the image at the right. Starting at the 0 degree/radian position (3 o'clock), the reading would be zero. Moving counter-clockwise motion to straight up (12 o'clock, 90 degrees, pi/2 radians), the y value would be 9.8. Another quarter turn (9 o'clock, 180 degrees, pi radians) would yield a zero value for y again. The bottom of the wheel (6 o'clock, 270 degrees, 3pi/2 radians) would be -9.8 m/s^2 before returning to the starting position (3 o'clock, 360 degrees, 2pi radians) to complete one revolution of the wheel. The resulting graph would be the same as 9.8 m/s^2 * sin(O).

Following the same pattern for the x values, at 0 radians, the x value is 9.8 m/s^2. At pi/2 radians, x is 0. At pi radians the x value is -9.8, 0 at 3pi/2 radians and finally back to 9.8 at 2pi radians. The graph is the same as 9.8*cos(O) which is shown at the left. Recall these graphs are those which would be expected with perfect placement of the accelerometer and no centripetal acceleration.

So what is the modulus? The modulus is the length of magnitude off the two vectors added together. Since the vectors are right angles, we may use the pythagorean theorem to find the magnitude of the sum of the vectors. Hence (9.8 cos x)² + (9.8 sin x)² = c². Manipulating the equation one may get 9.8² cos²x + 9.8² sin² x = c² or 9.8² (cos² x + sin² x) = c². Since cos² x + sin² x = 1, we can substitute and get 9.8² = c² or c = 9.8. As a result, one would expect the magnitude to always be 9.8 m/s^2 as you move around the circle if no centripetal acceleration is present. However, with centripetal acceleration, the acceleration is no longer constant.

The graph at the right shows the modulus in green. The accelerations experienced on the ride were between near 0 g to slightly more than 3 g's. Interestingly, the black curve is the x-axis of the accelerometer. It is much like was predicted in the discussion above, ranging from -9.8 m/s^2 to 9.8 m/s^2. However, the y-axis shown in purple does not appear similar to what was discussed previously. Centripetal acceleration produces an acceleration toward the center of the circle. In mathematical terms, the constant centripetal acceleration produces a vertical shift on the graph.

These concepts are not easily understood and are difficult to explain as you will note the whiteboard presentations being discussed at the left. If one looks downloads the data in text format, TI-83 format or Graphical Analysis format, you will also see that the z-axis which is positioned laterally is not zero as would be expected. The conclusion to be drawn from this is that the accelerometer was not positioned perfectly on the individual as the ride moved through its vertical circle. If you download the data, can you fit a mathematical model to the data?

Sea Dragon

Merged Data

Because of the inability to collect complete sets of data for some amusement park attractions with the equipment available to us, the entire class was organized to collect data over an entire ridel. We chose a roller coaster, The Mamba, to collect data on. One group ran a data set over the entire ride with very large intervals between data points. Other groups chose particular portions of the ride to collect data. Together, the entire class mapped the entire ride in greater detail. The data sets are shown below which the group put together.

A data set for the Mamba at 0.5 second intervals for the entire ride. The ride consists of being towed to the top of the first hill and dropping into the first valley before climbing back to the first hill and falling back into a second valley. The ride then climbs a shorter hill and proceeds down into a 540 degree turn before entering a shorts set of brakes and 5 camel back humps. The data set at the left is available in text format, TI-83 format and Graphical Analysis format. Note that this data has the X-axis pointing backwards hence acceleration forward will be shown as a negative value.

Mamba from top of hill 1 through the first valley into hill 2. The green points are the modulus of the three axes. The black set of data is the X-axis which was pointed backwards and is therefore negative for forward motion. The purple data set is the vertical axis. Data sets: Text TI-83 Graphical Analysis

Mamba from top of valley 1 over the top of hill 2 and into second valley. The green points are the modulus of the three axes. The black set of data is the X-axis which was pointed backwards and is therefore negative for forward motion. The purple data set is the vertical axis. Data sets: Text TI-83 Graphical Analysis

Mamba from second hill into the second valley and up to hill 3 which leads to the 580 degree turn. The green points are the modulus of the three axes. The black set of data is the X-axis which was pointed backwards and is therefore negative for forward motion. The purple data set is the vertical axis. Data sets: Text TI-83 Graphical Analysis

Mamba from the top of hill 3 into the 580 degree turn which has acceleration inall three axes but is dominated by the lateral acceleration measured by the z axis and shown in red on the graph. As with previous graphs, he green points are the modulus of the three axes. Data sets: Text TI-83 Graphical Analysis

Mamba through the camel humps. The black set of data is the X-axis which was pointed backwards and is therefore negative for forward motion. The purple data set is the vertical axis. Data sets: Text TI-83 Graphical Analysis

Ripcord

The final section of the data which students analyzed was the Ripcord. The Ripcord is further described in its own section.

Reaction Paper

Towards the end of the unit, students were asked to write a reaction paper. In this paper, students were asked to discuss things which they learned through the amusement park project, things which made the project useful, things which could have made the project more meaningful or created fewer problems for the students. Below is a sampling of student responses.

	"I learned that physics and math effects our every day lives. From the simple machines (i.e. inclined plane, pulley) to complicated machines (i.e. bungee jumping, roller coasters) the laws of physics and math apply to everything."
	"Our discussion in class was probably the best discussion we had all year. I really understood what people were talking about, with the knowledge that I have gained all year. It made a lot of sense to me and with everyone explaining their data it helped me understand what was really going on with the rides. Hopefully our final will make just as much sense to me, and I will be able to explain it as well as the worlds of fun lab."
	"We should have checked better here in class before going to the park whether our calculators, CBL's, accelerometers, and other equipments (sic) were in good condition and whether we had calibrated it right or not. Due to bad calibration, none of our data worked while we spent so much time just to get them right. It is totaly (sic) our fault that this happened but I am sure it would have turned out good if we had checked our calibration beforehand more than once."
	"Although the whiteboards would have been a great review for a formal final, the true test will be applying the concepts learned and reviewed to a new set of data."
	"I learned that data is collected better when the motion is fluent and not jerky. Jerky motion made graphs hard to understand and it was hard to find the motion one was looking for. I also learned that the location of the acclerometer on the pack affected the results of a given axis but did not affect the results of the fifth list (modulus)."
	"A major problem was that some people were reliant upon others to run the technology, since they did not know what to do or how to do it, even though it had been discussed."
	"My understanding of how to read acceleration graphs has improved imensily (sic) since we started this unit.... Before, when Iooked at an acceleration graph I saw confusion. Now when I look at an acceleration graph, I see something accelerating, doing a flip, jumping, and so on. ...this unit has helped me understand certain concepts of physics and pre-calculus that I never would have understood without applying these concepts to something real."

Final Exam

The final exam consisted of a a random group of 3 students preparing and presenting a random data set. The data sets were all collected prior to the final and were similar to what would be expected from an amusement park ride. The data sets varied from the accelerometer being attached to a spinning wheel (horizontal, vertical, at an angle), a spring oscillating back and forth, and an accelerometer attached to a pendulum hung over a pulley. As the pendulum swung back and forth, the length of the pendulum was decreased by pulling the string over the pulley. Each group was provided

• a data set

• a picture of the apparatus used to collect the data

• specific measurements related to the data collection such as length, radius etc

During the double period final exam, students were given the first period to analyze the data and prepare a presentation. During the second period, each group was given a specific amount of time to make a presentation. Questions and comments were accepted from the other members of the class. Each teacher scored each student based upon:

• work in the group to prepare the presentation

• knowledge dispalyed and quality of the student's portion of the presentation

• questions/comments while other groups were making presentations

More information regarding the final exam is available.

Amusement Home | Lesson Plan | Planning | The Trip | Ripcord | Assessment | Final Exam