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. XXXXXXXXXX (Bonus) Introduction □ Motivation – reason for your report (1) □ Hypothesis – what do you expect to see, and why? (1) XXXXXXXXXX Rigid Pole Model □ Correct polynomial fit for take-off velocity (1) □ Plot for rigid pole model (1) □ Maximum height + discussion (2) XXXXXXXXXX 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) XXXXXXXXXX XXXXXXXXXX Flexible Pole Model □ Derivation of flexible pole model formula (2) □ Plot of flexible pole model (1) □ Maximum height (1) □ Discussion and model comparison (3) XXXXXXXXXX 3 XXXXXXXXXX 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) XXXXXXXXXX 3 XXXXXXXXXX Conclusions □ Summary of findings (1) □ Inclusion of all quantitative results (1) □ Suggestions for further model improvements (1) XXXXXXXXXX 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) XXXXXXXXXX Style and creativity □ Presentation and neatness – including plots, formulas, and section headings (2) □ Organization, flow, grammar, and error checking (1) □ Presentation of plots (1) XXXXXXXXXX 0 Up to 2 bonus marks for outstanding plots/formatting Name: XXXXXXXXXXTotal / 35 IPS1500 Case Study - Pole VaultUniversity of Guelph — 2021IntroductionThis case studywill introduce students tomodeling in physics. At its core, physics is aboutbuilding models to represent physical phenomena into predictable and reproducible re-sults. Models start with the most significant variables, and are further refined by takingother relevant but less important variables into account. In this case study we will beinvestigating the pole vault. The pole in the pole vault allows the vaulter to convert theirkinetic energy in the run-up into potential energy at the apex of their jump to pass overa bar.1 Rigid Pole ModelWe will first model the pole vaulter as a point mass on the end of a rigid pole.Figure 1: Schematic of the pole vault setupFigure 1 shows a pole vaulter at take-off and the apex of their jump. A pole vaulter’svelocity v at take-off is determined by their take-off angle, φ, as shown in Figure 2.1Info: Since our model will only focus on the energy converted from the run-up intothe height at the vaulter’s apex, We will not be modeling the work the vaulter doesduring the jump. To remedy this, add 0.80 meters to any calculated grip height toaccount for the work done by the vaulter to push off of the pole at the apex of theirvault. Additionally, subtract 0.20 meters from any grip height to account for thedepth of the take-off box where the pole is planted.iFigure 2: Take-off velocity v as a function of take-off angle φWe will start by assuming all energy is conserved during the take-off and jump, i.e.12Mv2 +Mghinitial = Mghfinal (1)Question 1A pole vaulter’s grip height is the distance from the bottom of the pole to their centreof mass. For the purposes of our model, this is the height hfinal. Qualitatively, what willhappen if a pole vaulter holds the pole at a height greater than hf inal when they takeoff?Question 2In the Jupyter Notebook “IPS1500 Case Study”, there are a few libraries and two arrayscontaining some data points extracted from Figure 2. Create a polynomial fit of this datausing np.polyfit() and np.poly1d(). Based on Figure 2, would a linear, quadratic or cubicfit best represent the data? What is your equation for the polynomial fit?Question 3Define 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 polynomial2function to determine the take-off velocity for each take-off angle and use the parametershinitial = 1.85m, M = 80kg, and g = 9.8m/s2. Why does a shallower take-off anglecorrespond to a greater take-off velocity? The world record for a pole vault is 6.16meters, is your model realistic? Why or why not?2 Energy LossIn reality, energy is not conserved during the take-off. Upon planting the pole, the polevaulter dissipates energy into their body depending on their velocity parallel to the pole.This dissipated energy is given by the function:∆E =12Mv2 cos2(φ+ α) (2)Question 4Using figure 1, determine the angle α in terms of the parameters hinitial = 1.85m andL0 = 5.0m. Insert equation (2) into equation (1) and simplify to get an expression forthe grip height.Question 5As in question 3, define a new function for the grip height and plot the grip height fortake-off angles between 0° and 90° taking the dissipated energy from equation (2) intoaccount. What take-off angle is optimal for the greatest height?Info: Python’s trig functions only perform calculations in radians. Make sure toconvert all of your angular inputs into rads before performing any operations.i3 Flexible Pole ModelFlexible 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 bythe equation:∆E =F 202kcos2(φ+ α) (3)where F0 is the Euler buckling load and k is a constant related to the vaulter’s ability toresist backwards forces when planting the pole. F0 is proportional to the stiffness ratingof a pole given by a manufacturer, and is directly related to how much compression forceis required to make a pole buckle.Question 6Using equations (3) and (1), define a function for the grip height and plot it for take-offangles 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?3Question 7Vary the buckling load value F0 and qualitatively describe the effect on vault height andexplain why this is the case. Is there a lower limit? What is the reason for the lowerlimit?4 Coaches BoxAfter clearing the bar, pole vaulters needs to land on a soft mat called the “pit". Themiddle of the pit is called the “coaches’ box", the centre of which is about 2.25 m behindthe bar.Question 8Assuming the safest place to land is the centre of the coaches’ box, we will impose aconstraint 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 horizontalat the apex of the jump.Question 9Using the velocity determined in question 8, determine the vaulter’s kinetic energy at theapex of their jump and re-calculate their grip height and vault height. Compare this valueto average values for Olympic male pole vaulters. How do they compare? Comparing thisvalue to those from section 3, why does a flexible pole allow for greater vault heights?Question 10The fastest average sprint speed achieved by a human is 10.43 ms−1 by Usain Bolt. Thefastest recorded instantaneous speed by a human sprinter is 12.1 ms−1 by DonovanBailey. What heights could a pole vaulter achieve if their run-up reached these speeds?Are they this feasible?5 ConclusionsYou will have no doubt noticed that we have made many generalizations and simplifica-tions in our model. How could it be further refined?4IPS*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