## Modelling Rocket Launch with Calculus

**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.

www.youtube.com/watch?v=TXMGu2d8c8g

**Possible Solution for task:**

**Introduction (formulate)**

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 |

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 |

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.