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.
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(Bonus)
Introduction
□ Motivation – reason for your report (1)
□ Hypothesis – what do you expect to see, and why? (1)
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Rigid Pole Model
□ Co
ect polynomial fit for take-off velocity (1)
□ Plot for rigid pole model (1)
□ Maximum height + discussion (2)
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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)
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Flexible Pole Model
□ Derivation of flexible pole model formula (2)
□ Plot of flexible pole model (1)
□ Maximum height (1)
□ Discussion and model comparison (3)
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3
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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)
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3
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Conclusions
□ Summary of findings (1)
□ Inclusion of all quantitative results (1)
□ Suggestions for further model improvements (1)
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Up to 1 bonus mark for
eally great ideas
Appendix
□ Appropriate use of appendix for derivations (1)
□ Code used in jupyter notebook to generate plots, etc (1)
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Style and creativity
□ Presentation and neatness – including plots, formulas, and section headings (2)
□ Organization, flow, grammar, and e
or checking (1)
□ Presentation of plots (1)
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0
Up to 2 bonus marks for
outstanding
plots/formatting
Name: XXXXXXXXXXTotal / 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
uilding 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 thei
kinetic energy in the run-up into potential energy at the apex of their jump to pass ove
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 thei
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 li
aries and two a
ays
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
co
espond 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 fo
the grip height.
Question 5
As in question 3, define a new function for the grip height and plot the grip height fo
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 ca
on 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
esist 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 lowe
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