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Roller Coasters provide a wonderful context for mathematical modelling. The actual construction of a roller coaster is a marvellous example of engineering, mathematics and physics. From a technology aspect students can get visual feedback by placing an image in the background of the graphing screen. Students can build an understanding of how the degree of a polynomial determines features such as the quantity of turning points, the nature of odd and even degree polynomials and curvature. In the example shown here a quartic function has been used to represent a section of the roller coaster. The x-axis was aligned approximately with the horizon and the image scaled using known information. The turning points on the function were used to model the peaks and troughs of the track which provided no freedom to adjust the curvature to align the remaining sections, even those within the domain of the model. Students are invited to consider other functions, including piecewise functions that may produce a more appropriate model with greater flexibility.

If an appropriate scale is set, students can also include calculations of: – Ride length – Gravitational potential energy – Kinetic energy – Velocity – Time

Students can also explore how concavity affects the ride experience. For an easier starting context, students can consider water slides that typically have downhill sections only. There are numerous water slides around the world worth studying. “Summit Plummet” at Disney’s Blizzard theme park in Florida is a relatively simple design. Consisting of a starting height of approximately 40 metres and an almost vertical drop, riders can achieve speeds of approximately 100km/h. The radar located at the base of the drop displays rider speeds and boasts the highest speed of the day. The following curves are all modelled on a drop of 40 metres over a horizontal distance of 40 meters. The blue line on each graph represents the shape of the water slide; the red dotted line is the speed at each point on the slide, assuming zero resistance.

It is clear to see that curvature has an enormous impact on the average speed. As we have assumed that there is no resistance to the rider’s motion, the maximum speed would be the same for each ride. The velocity is calculated based on the assumption that all the gravitational potential energy is transferred to kinetic energy, when friction is taken into account this is not true. It is easy to incorporate resistance into these types of models, particularly where a program is being used. A very simplistic approach would be to use:

In reality there are many complications that include the volume of water flowing, the weight and dimensions of the rider, the type and area of the bathing costume (nylon has a lower drag) and indeed, the amount of sunscreen the rider is wearing! It is interesting to note that increasing the volume of water slows down heavier riders but accelerates lighter riders, so the water acts as a moderating effect. Rider speeds on Summit Plummet typically vary between 80km/h and 110km/h. When piecewise functions are used to model any of these types of rides, students need to ensure that curves are joined smoothly or at least understand that this presents limitations to their model. Even if students have not started calculus, they should already understand the concept of gradient, technology can be used to determine the gradient where two curves meet.

Wine Glasses Wine glasses come in a range of shapes and sizes depending on their purpose. The tall stem is so that you don’t have to place your hands on the body of the glass, this is to avoid heating the wine with your hands. Red wine glasses tend to have a wider body to help the wine breathe and a relatively narrow rim to allow the concentrated aromatic flavours to rise and tantalise the pallet. The glass must also be designed to hold a standard drink (175ml for wine) and preferably at the widest point of the body. The rim to body area ratio should be around 1 : 1.5. Finally, stability plays a role in the design of the glass. If the stem is too long or the base too small the glass will be unstable and tip over too easily. There is a lot more to designing a good wine glass than first meets the eye. Considering some bottles of wine sell for $100.00’s and more, it makes sense to design the perfect glass from which to serve the wine. Students can design the wine glass to meet all the specifications, starting with a standard wine glass and using functions to model the curvature. To ensure the curves used to model the glass profile are functions, tip the glass on its side.

Mathematics: – Standard drink = 175ml (wine) – This should occur at the widest section of the glass (turning point) – Ratio between y ordinates at the turning points and top of the glass can be used to help match the rim to body ratio. – What is the ideal stem length? (Now it’s a STEM activity!) A piecewise function could be used to ensure the stem and vessel curves join smoothly. – To increase the complexity of the task the thickness of the glass could also be included in the modelling. Is it okay to simply translate the function modelling the outside of the glass or does this pose a problem with regards to glass thickness?

A range of calculus concepts can be dealt with in this task. If you are really adventurous, try using a 3D printer to produce a real model of your perfect wine glass.

This Activity was developed by T-Cubed Trainers Bozenna Graham and Stephen Broderick as part of their session What’s your Vector Victor? – Flying High with PSMT & TI-Nspire™ that they presented at the Brisbane Learn, Energise, Connect PD Day last November.

Context

Telemetry video data for SpaceX launches are readily available on the web. Analysis of this data verifies a number of different elements for the various stages of a launch such as acceleration, altitude, distance travelled and average speed. SpaceX has successfully launched a number of payloads into orbit including satellites and supplies to the International Space Station.

Task: Investigating a SpaceX rocket launch

Collect speed, altitude and time data for the first stage of the SpaceX launch contained in the stimulus link below. The first stage involves the time that the three rocket engines are firing. The end of the first stage occurs just before the rocket boosters are detached. You can add time data by pausing the video and noting the timestamp in the telemetry data. Use regression analysis to determine equations for Stage 1 and Stage 2 burns of the launch. Stage two occurs sometime after the 27-minute mark. Use calculus techniques to investigate the mathematical models produced from the data. Compare the acceleration and average speed during the Stage 1 and Stage 2 burns and also use calculus to determine the altitude of the SpaceX rocket shortly after launch.

To complete this task

• use the problem-solving and mathematical modelling approach to develop your response

• respond with a range of understanding and skills, such as using mathematical language, appropriate calculations, tables of data, graphs and diagrams

• provide a response that highlights the real-life application of mathematics

• respond using a written report format that can be read and interpreted independently of the instrument task sheet

• develop a unique response

• use both analytic procedures and technology.

Stimulus

Below is a Youtube link for a SpaceX launch. The telemetry data for speed, altitude and time is located on the bottom of the screen.

This task involves collecting SpaceX launch data during Stage 1 and Stage 2 of a launch.

The data collected includes time, altitude and speed of the SpaceX rocket. The starting times and duration of each stage will need to be determined from the video. Mathematical models for Stage 1 and Stage 2 burns will be determined and analysed with calculus techniques to determine the acceleration, distance travelled and average speed in each stage. The altitude in the early stages of the launch will also be approximated.

Results (Solve)

Some of the initial assumptions include:

Stage 1 involves the time when the three rocket boosters are firing and extends from t = 0 to t = 156 seconds

Stage 2 commences around 27^{1}/_{2 }minutes into the launch and lasts approximately 85 seconds

Regression analysis will be used to develop mathematical models for both stages

Altitude and distance travelled are only the same in the initial stages of a launch when the SpaceX rocket is travelling vertically

Distance travelled by the SpaceX rocket can be approximated with calculus techniques

Since this is actual data, atmospheric friction (or wind resistance) is included in the mathematical models

The telemetry data was collected from the video.

Table 1 includes the telemetry data for the first 156 seconds of the Stage 1 burn and includes time, speed (kilometres/hour) and altitude (kilometres). The speed in metres/second was added to the table by multiplying by the conversion factor (1000/3600)

Stage 1 Burn data

Time (seconds)

Speed (kilometres/hour)

Altitude(kilometres)

Speed(metres/second)

8

110

0.1

30.556

10

155

0.2

43.056

12

199

0.3

55.278

13

239

0.4

66.389

16

303

0.6

84.167

19

360

0.8

100

21

388

1

107.778

22

410

1.1

113.889

25

476

1.6

132.222

28

522

1.9

145

31

575

2.3

159.722

33

625

2.8

173.611

35

669

3.1

185.833

38

733

3.7

203.611

40

769

4

213.611

43

836

4.7

232.222

47

898

5.6

249.444

50

934

6.4

259.444

53

977

7.3

271.389

57

1021

8.2

283.611

60

1068

9.1

296.667

70

1326

12.2

368.333

90

2019

19.9

560.833

110

2943

30

817.5

130

4105

41.9

1140.278

150

5498

55

1527.222

156

5861

59.4

1628.056

Table 1: Stage 1 data for the SpaceX launch

The graph of time (seconds) versus speed (metres/second) for the Stage 1 burn in figure 1 is best represented by the cubic function below:

The shape of the graph suggests that the acceleration of the SpaceX rocket is increasing over the 156 second interval. The derivative of the speed function yields the acceleration function.

The reason for this difference is that the SpaceX rocket is not flying on a vertical trajectory at this stage; it has changed its pitch and is flying on an acute angle with the Earth’s surface. In actual fact, 81.258 km is the total distance that the SpaceX rocket has flown after 150 seconds.

The total distance travelled after the Stage 1 burn is illustrated in figure 4.

Stage 2 Burn data

The data in Table 2 was collected after 27:37 minutes of flight time. The second stage burn lasts for approximately 85 seconds.

Time (seconds)

Speed (km/h)

Altitude (km)

Speed (m/sec)

0

26572

198

7381.111

10

27377

199

7604.722

20

28459

200

7905.278

30

29419

201

8171.944

40

30428

203

8452.222

50

31437

205

8732.5

60

32760

208

9100

70

34157

212

9488.056

80

35647

216

9901.944

85

36694

221

10192.778

Table 2: Stage 2 data for the SpaceX launch

The graph for the 85-second burn during Stage 2 starts after 27 minutes and 37 seconds into the launch and is shown below in Figure 5.

The data can be represented as a linear model. The correlation coefficient for the association is 0.99417, which means that the algebraic model is a close match to the empirical data.

The distance travelled (Figure 6) in the Stage 2 burn of 85 seconds is approximately 734 kilometres.

Conclusion

The trajectory of a SpaceX rocket varies considerably during a launch. In this task different mathematical models were used to represent Stage 1 and Stage 2 of the launch. A cubic equation with a correlation coefficient (r ) of 0.999537 was used to model Stage 1, while a linear equation with a correlation coefficient (r) of 0.99417 was used to model Stage 2.

The acceleration rates for Stage 1 and Stage 2 are quite different due to the effects of gravity. For Stage 1, the acceleration ranges between 4.79 m/sec^{2} and 23.99 m/sec^{2}, whereas for Stage 2, acceleration is approximately 32.51 m/sec^{2} throughout the 85 second burn. This difference is due to the effects of gravity. As the SpaceX rocket gets further away from Earth, the effects of gravity decrease by a factor of 1/distance^{2} in accordance with the inverse-square law. This results in greater acceleration of the SpaceX rocket as it escapes the Earth’s gravity. The integration of the mathematical model for stage 1 can be used to determine the altitude when the rocket is launching vertically. There is a good agreement with the telemetry data for up to 40 seconds of the Stage 1 burn; however, after this period of time, the rocket’s trajectory follows a curved path making the determination of the altitude difficult. The integration of the models can be used to determine the total distance travelled during the Stage 1 and Stage 2 burn. The distances travelled by the SpaceX rocket in Stage 1 and Stage 2 are 90.744 and 733.383 km respectively. During Stage 2, the rocket travels 8 times further than it does in Stage 1 and in half the time. The average speeds for Stage 1 and Stage 2 are 0.5817 and 8.628 km/sec respectively. The SpaceX rocket is travelling nearly 15 times faster in Stage 2 than it is in Stage 1. Although average speeds are useful for looking at and comparing various stages of a launch, precise speeds are needed for docking with the International Space Station which travels at 7.9 km/sec.