Roller Coaster MA131 PROJECT Part 2: Adding a camera

Background and problem setting

The CEO liked your original submission, though she did return it with a few recommendations for improvement additionally, she has requested that you make an addition to that submission (as a new section to the overall document).

The client would like you to place a camera somewhere over the roller coaster that would be in a fixed position but would record video of a rider during some portion of the ride. In addition to de- ciding on the placement of the camera, you will need to determine the specification for camera rotation rate that will be required so that the camera can follow the rider. (Some additional details provided below.)

Design Goal restated

The CEO has asked you to build the 2-d profile of a section of track that will be part of a new coaster. The theme park that has commis- sion the design has already decided that the start of the climb will rise to a height of 70 m. Your job is to design what happens over the next 160 m, measured horizontally. (See Figure 1). Additionally, the CEO, a bit of a math enthusiast, has asked that you meet the follow- ing conditions on the design.

Make it fun.

The height h must be a function of horizontal position x.

h(x) can be piecewise defined, but she wants to have a formula for each section of the track.

No digging the track must not go below ground.

Physics dictates that no portion of the track should go above 70 m, or the coaster wont work.

Added Requirements:

Place a camera somewhere over the track such that the camera can record at least 1.0 seconds of video on a rider. The camera will have a narrow (zoomed in) field of view, so it will need to track the motion of the coaster.

The intention is not for you to write an entirely new project. Rather, you will be revising your first project write- up, correcting any errors, and simply adding a section or two to address this camera requirement.

Figure 1: Your section of the coaster starts at (0, 70) and ends at x = 160. The rise up to 70 m is not your respon- sibility.

Estimating Construction costs

The CEO has directed that you also provide an estimate of the con- struction cost for your section of the coaster, where your companys methodology for this estimate requires you to account for three fac- tors in performing that estimate.

1. The track does not simply float in the air, but is supported by a trestle structure. The amount of support structure required is roughly proportional to the crosssectional area A under the track. So, this material cost is estimated as

Cm =c1A, (1)

where c1 = $10000.

The actual value for Cm will also depend upon several other engi- neering factors which cannot be estimated until the next phase of the design process (when engineering material specifications are determined). So, the CEO has asked that your estimate for Cm sim- ply needs to be a rough estimate, but it needs to be an overestimate of the actual area.

2. The vertical rises and falls of the track make the track longer and add to the production and labor costs. If Tf is the total measure of all of the drops in the track (measured in meters), and Tr is the total amount of rises (in meters) for your portion of the design, then the associated cost is given by

Ct =c2(160+Tf +Tr), (2)

where c2 = $60000.

3. The hazard of building on highly sloped track requires a labor surcharge associated to additional safety requirements during the construction process, estimated as

Csafety =c3m, (3)

where c3 = $500, 000, and m is the absolute value of the steepest slope on the track. However, there is an upper limit of $2, 000, 000 associated to this safety surcharge, so that

Cl = Csafety Csafety < $2,000,000 (4)

$2, 000, 000 else.

4. The total cost estimate is then given by C = Cm + Ct + Cl . (5)

Figure 2: To get a basic material cost, you will need to estimate the area under the curve for the coaster that you design.

Figure 3: The design shown has a 60 m drop, a 40 m rise, then a 50 m drop, for a total vertical change of 150 m.

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Thrill Factor

There are many different ways to evaluate how fun a ride will be. For roller coasters, one of the standard calculations is the thrill factor for a vertical drop, defined as the product of the angle (mea- sured in radians) of the steepest part of in the descent on the drop and the total vertical distance of that drop.

The CEO would like you to assess the thrill factor associated with your segment of the coaster.

As a means to assess the efficiency of your design, the CEO also wants you to report the thrill per dollar for your segment design.

New design component adding a camera.Some preliminary physics.

A roller coaster coasts. All the energy is provided from the climb up the first hill. As the cars are pulled to the top of the first hill, they gain potential energy. When allowed to roll down, the potential energy is converted to kinetic energy and when they coast uphill, the reverse occurs, with cars slowing as they gain height. For this project, we will assume that friction is small enough that we may discount its effects.The potential energy is given by

P = mgh,

where m is mass (kilograms), g is the gravitational acceleration (me- ters/second/second), and h is the height in meters above some ref- erence. The resultant energy is measured in Joules (J). Kinetic energy (energy due to the speed of an object with momentum) is given by

K = 1mv2, 2

where v is the magnitude of velocity, in meters/second. At any point along the track, the maximum possible speed achieved would occur if all of the potential energy is converted to kinetic (which ignores the effect of friction):

m g h = 1 m v 2m a x , 2

where h is the vertical distance between the top of the track and the point of interest.

Design Objective Camera system specifications.

The video camera that must follow the coaster to take a short video clip of each ride. The camera can be mounted anywhere over the

As a roller coaster designer, you (of course) already know these physics. However, as a calculus student, you might not. So we have included a short explanation about how you compute velocities on your coaster.

roller coaster ma131 project part 2: adding a camera 3

track (using cables). The idea is to mount the camera somewhere over the track such that it can capture an interesting part of the ride (so that we can sell the video to the rider). As an illustration of de- ployment of such a system on a previously designed track, take a look at Fig. 4.

Your report should include a specification for the camera mount- ing system, specifying the rotational speed (angular velocity) in radians per second that is required to track the coaster on the por- tion of the track where you intend to take the video. This means that you need to determine the rate of change of (t) and when the coaster is at various points on the track (you should be able to do that using the speed of the coaster, v(t). The speed can be computed using the energy conservation equation

mgh = 1/2mv2.Revised Project Deliverable

The CEO1 has tasked you with using your creative flare to develop and analyze a design and to prepare a Technical Report describing the design and your assessment of that design. Include a section de- scribing your camera specification (to include appropriate supporting description.)

Additional supporting guidance

You are required to submit a typewritten paper that describes your solution. You should use the handout on technical writing to help

Figure 4: Schematic. This illustration came from a coaster we designed a few years back. The camera must rotate at suffi- cient speed to be able to track (t) as the coaster moves along the track. Ob- viously, your track design and camera placement will be different.

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Automated Camera

Note this element defines the major new mathematical work required in this project. You get to choose camera placement, but then you will need to do a related rates problem to compute the rotational rate ((t) that your camera must achieve.

1 As a starting point, use your original submission. Make any recommended changes (based on feedback) and check your computations. Add the new section on the camera. You should not start this project from scratch, as the CEO has already reviewed your initial draft.

you compose your project writeup.2 It is essential that you keep in mind the following guidelines while preparing your manuscript:

Your manuscript must stand on its own. Do not write as if your reader had a copy of this project or even that they fully un- derstand the questions asked. Your audience is the CEO, who understands math and roller coaster design, but will not be famil- iar with the details of your project (as she is monitoring several such projects.)

Technical writing (and business writing) is not like creative writing and story telling. Make sure you read the writing guide to help you understand the difference.

Your paper should have sections composed of paragraphs made from sentences. It is meant to be read. It should not be a list of equations and numbers.

Collaboration and Academic Integrity.

Clarkson University Regulations provide general guidance: The Clarkson student will not present, as his or her own, the work of another, or any work that has not been honestly performed, will not take any examination by improper means, and will not aid and abet another in any dishonesty. The following additional guidance is provided:

You do this project as a team, but your team must be formed from students who have the same lecture instructor. Your instructor will provide additional details regarding team composition.

You may not discuss the problem with ANY LIVE PERSON exceptyour partner, MA131 TAs, or Math Department instructors.

You may discuss non-problem issues, (such as formatting, how to use Word Equation Editor, or how to make plots using Desmos of Excel) with anyone, but you must be asking general usage ques- tions, not project problem questions. If in doubt, ask a TA or an instructor. Additionally, you may go to the writing center for help on style and grammar aspects.

If a TA or an instructor helps you on the problem, you must in- clude an acknowledgment in your project.

You may use your texts freely. However, you should include them in your list of references if you use them.

2 Material is posted on Moodle to assist you in understanding how to write a mathematical paper.

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Your track design will likely involve the stitching together of more than one function a piecewise defined track. Desmos provides an excellent way for you to visualize and experiment with possible designs.

You may use other references, both from the library and on line, but you must use appropriate citations where you use that mate- rial. Remember, general mathematical knowledge does not need to have a citation, but if you have to look up an equation or formula, your general practice should be to cite that reference.