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IPS*1500 – Case Study Assignment – The Pole Vault– Marking Scheme Section Mark Early Bird Sections 1 & 2 completed, including Python code, and handed in (through the Courselink dropbox) no later than 11:59 pm, Friday, November 5th. 2 1 0 (Bonus) Introduction □ Motivation – reason for your report (1) □ Hypothesis – what do you expect to see, and why? (1) 2 1 0 Rigid Pole Model □ Correct polynomial fit for take-off velocity (1) □ Plot for rigid pole model (1) □ Maximum height + discussion (2) 4 3 2 1 0 Effect of Energy Loss □ Derivation of energy loss formula and Alpha (2) □ Plot for energy loss model (1) □ Maximum height (1) □ Discussion and comparisons between models (2) 6 5 4 3 2 1 0 Flexible Pole Model □ Derivation of flexible pole model formula (2) □ Plot of flexible pole model (1) □ Maximum height (1) □ Discussion and model comparison (3) 7 6 5 4 3 2 1 0 Flexible Pole Model With Constraints □ Derivation of v and constraint implementation (2) □ Plot of flexible pole model with constraints (1) □ Maximum height (1) □ Discussion and model comparison (3) 7 6 5 4 3 2 1 0 Conclusions □ Summary of findings (1) □ Inclusion of all quantitative results (1) □ Suggestions for further model improvements (1) 3 2 1 0 Up to 1 bonus mark for really great ideas Appendix □ Appropriate use of appendix for derivations (1) □ Code used in jupyter notebook to generate plots, etc (1) 2 1 0 Style and creativity □ Presentation and neatness – including plots, formulas, and section headings (2) □ Organization, flow, grammar, and error checking (1) □ Presentation of plots (1) 4 3 2 1 0 Up to 2 bonus marks for outstanding plots/formatting Name: Total / 35 IPS1500 Case Study - Pole Vault University of Guelph — 2021 Introduction This case studywill introduce students tomodeling in physics. At its core, physics is about building models to represent physical phenomena into predictable and reproducible re- sults. Models start with the most significant variables, and are further refined by taking other relevant but less important variables into account. In this case study we will be investigating the pole vault. The pole in the pole vault allows the vaulter to convert their kinetic energy in the run-up into potential energy at the apex of their jump to pass over a bar. 1 Rigid Pole Model We will first model the pole vaulter as a point mass on the end of a rigid pole. Figure 1: Schematic of the pole vault setup Figure 1 shows a pole vaulter at take-off and the apex of their jump. A pole vaulter’s velocity v at take-off is determined by their take-off angle, φ, as shown in Figure 2. 1 Info: Since our model will only focus on the energy converted from the run-up into the height at the vaulter’s apex, We will not be modeling the work the vaulter does during the jump. To remedy this, add 0.80 meters to any calculated grip height to account for the work done by the vaulter to push off of the pole at the apex of their vault. Additionally, subtract 0.20 meters from any grip height to account for the depth of the take-off box where the pole is planted. i Figure 2: Take-off velocity v as a function of take-off angle φ We will start by assuming all energy is conserved during the take-off and jump, i.e. 1 2 Mv2 +Mghinitial = Mghfinal (1) Question 1 A pole vaulter’s grip height is the distance from the bottom of the pole to their centre of mass. For the purposes of our model, this is the height hfinal. Qualitatively, what will happen if a pole vaulter holds the pole at a height greater than hf inal when they take off? Question 2 In the Jupyter Notebook “IPS1500 Case Study”, there are a few libraries and two arrays containing some data points extracted from Figure 2. Create a polynomial fit of this data using np.polyfit() and np.poly1d(). Based on Figure 2, would a linear, quadratic or cubic fit best represent the data? What is your equation for the polynomial fit? Question 3 Define a function for the pole vaulter’s grip height and plot the grip height for take- off angles between 0° and 90° using at least 50 different points. Use your polynomial 2 function to determine the take-off velocity for each take-off angle and use the parameters hinitial = 1.85m, M = 80kg, and g = 9.8m/s2. Why does a shallower take-off angle correspond to a greater take-off velocity? The world record for a pole vault is 6.16 meters, is your model realistic? Why or why not? 2 Energy Loss In reality, energy is not conserved during the take-off. Upon planting the pole, the pole vaulter dissipates energy into their body depending on their velocity parallel to the pole. This dissipated energy is given by the function: ∆E = 1 2 Mv2 cos2(φ+ α) (2) Question 4 Using figure 1, determine the angle α in terms of the parameters hinitial = 1.85m and L0 = 5.0m. Insert equation (2) into equation (1) and simplify to get an expression for the grip height. Question 5 As in question 3, define a new function for the grip height and plot the grip height for take-off angles between 0° and 90° taking the dissipated energy from equation (2) into account. What take-off angle is optimal for the greatest height? Info: Python’s trig functions only perform calculations in radians. Make sure to convert all of your angular inputs into rads before performing any operations. i 3 Flexible Pole Model Flexible fiberglass or carbon fiber poles have been used in pole vaulting since the 1960s, and have allowed vaulters to clear greater heights than with rigid steel or bamboo poles. The energy dissipated in the vaulter’s body at take-off using a flexible pole is given by the equation: ∆E = F 20 2k cos2(φ+ α) (3) where F0 is the Euler buckling load and k is a constant related to the vaulter’s ability to resist backwards forces when planting the pole. F0 is proportional to the stiffness rating of a pole given by a manufacturer, and is directly related to how much compression force is required to make a pole buckle. Question 6 Using equations (3) and (1), define a function for the grip height and plot it for take-off angles between 0°and 90°. Use the parameters hinitial = 1.85m, M = 80kg, L0 = 5.0m, F0 = 800N , k = 250Nm−1 and g = 9.8m/s2. What is the greatest vault height? 3 Question 7 Vary the buckling load value F0 and qualitatively describe the effect on vault height and explain why this is the case. Is there a lower limit? What is the reason for the lower limit? 4 Coaches Box After clearing the bar, pole vaulters needs to land on a soft mat called the “pit". The middle of the pit is called the “coaches’ box", the centre of which is about 2.25 m behind the bar. Question 8 Assuming the safest place to land is the centre of the coaches’ box, we will impose a constraint so that the vaulter will land there after clearing the bar. Using kinematics, determine the velocity at the top of a 5.6 meter vault. Assume all velocity is horizontal at the apex of the jump. Question 9 Using the velocity determined in question 8, determine the vaulter’s kinetic energy at the apex of their jump and re-calculate their grip height and vault height. Compare this value to average values for Olympic male pole vaulters. How do they compare? Comparing this value to those from section 3, why does a flexible pole allow for greater vault heights? Question 10 The fastest average sprint speed achieved by a human is 10.43 ms−1 by Usain Bolt. The fastest recorded instantaneous speed by a human sprinter is 12.1 ms−1 by Donovan Bailey. What heights could a pole vaulter achieve if their run-up reached these speeds? Are they this feasible? 5 Conclusions You will have no doubt noticed that we have made many generalizations and simplifica- tions in our model. How could it be further refined? 4 IPS*1500 Case Study - The Pole Vault – Guidelines Your final Case Study report is due Friday, November 19th at the start of class. You should submit a physical copy of your report, as well as a Dropbox submission. General Guidelines: The case study, including any equations, should be typed. You may discuss how to do the calculations with your classmates, but you must do the work yourself. Do not, under any circumstances, send a copy (digital or paper) of any part of your report to a classmate - there have been too many occasions where this resulted in copying sections (directly or indirectly), which is plagiarism and will cause both of you to lose marks whether or not copying was your intention. Your report should not be a set of disjointed calculations - you should have text throughout the report explaining what you are doing. Do not simply address the questions in the case study in point form: your answers should be incorporated into the text. You do not need to point out where the questions are answered, we will find