Skyline - The Big Sky Undergraduate Journal
Volume 1 | Issue 1
Article 12
2013
Predicting the Pole Vault
Caitlin Maulin
Idaho State University, [email protected]
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Maulin, Caitlin (2013) "Predicting the Pole Vault," Skyline - The Big Sky Undergraduate Journal: Vol. 1 : Iss. 1 , Article 12.
Available at: http://skyline.bigskyconf.com/journal/vol1/iss1/12
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Predicting the Pole Vault
Acknowledgments
Faculty Mentor - Charles Scott Benson Jr., Ph.D., Department of Economics
This research article is available in Skyline - The Big Sky Undergraduate Journal: http://skyline.bigskyconf.com/journal/vol1/iss1/12
Maulin: Predicting the Pole Vault
Abstract
The purpose of this study was to take data that was collected by Dave Nielsen, Head
Track Coach at Idaho State University, and conduct a regression analysis that results in an
equation that can be used as a future training tool for the pole vault. The data was collected using
DartFish software to analyze videos that were taken at 8 different track meets in 2004, including
the U.S. Olympic Trials and the Reno Pole Vault Summit. The findings of this study show
empirically that the hypothesized physics on proper vaulting technique is consistent with results
of the regression analysis.
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Introduction
The pole vault is a mystifying sport in which an athlete runs down a runway with a pole,
plants the pole into a metal box in the ground, swings upside down with the hope of clearing a
bar, and lands on a soft pit. The bar is set on an apparatus called the standards, which have the
ability to be moved closer or farther away from the front of the pit. Pole vaulting has been the
focus of physics analysis in order to understand how to create the most successful, efficient
vault. When pole vault was a new sport, people used bamboo and steel poles which greatly
decreased an individual’s ability to clear high bars. Today’s pole vaulters use carbon fiber and
fiber glass poles which have significantly improved maximum potential heights that can be
cleared. The world record clearance for men is 6.14 meters (20 feet 1.73 inches), and for women
is 5.06 meters (16 feet 7.22 inches) (IAAF Athletics). Modern pole vaulting poles are designed to
store strain energy, which is defined as energy stored in a solid due to deformation (Frèrea,
L'Hermette, Slawinskib, and Tourny-Chollet 123-138). For a vaulter to reach their maximum
height potential, the run-up velocity (kinetic energy) needs to be stored in the pole as strain
energy, and then converted into gravitational potential energy of the vaulter (Fellah Jahromi, Xie,
B. Bhat, and Atia 41-53).
According to current literature by Dave Nielsen, Dr. Peter McGinnis, and others there is a
general consensus on the important aspects of the vault. Each study notes the importance of the
run up before the vault. These studies also note the benefit of having the “tallest” takeoff. The
more upright the vaulter is at takeoff the greater the potential energy is. The main goal for a
vaulter is to maximize potential and kinetic energy at takeoff, as well as minimize energy lost
during the actual vault (McGinnis). In the following sections, data was taken from elite vaulters
at 8 different meets during 2004. The data measures different parts of their vaults and
corresponds with different transfers of energy throughout the vault. The model tests empirically
if these conclusions from previous studies are consistent in a real world setting.
Data
Originally there were 849 observations. After analysis, the sample size was pared down
to 522 observations with complete data for all of the variables. Various reasons may cause one or
more parts of the data to be insufficient such as a vaulter not carrying out the entire vault, or
technical issues with the camera.
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Maulin: Predicting the Pole Vault
Variables
The variables used in the analysis were collected by Dave Nielsen, and are listed below.
SUCCESS- This variable takes the value 1 when a vaulter clears the bar, and takes the value 0 if
the vaulter misses the bar.
HEIGHT- The bar the vaulter jumps over is set in meters. A general progression from starting
height is 10 to 15 centimeters. The expected sign for this coefficient is negative because as the
bar increases in height so does the difficulty.
VELOCITY- Velocity is found by dividing five meters by the time it takes a vaulter to run the
last five meters of their approach, which is marked on the runway. Velocity is measured in
meters per second (m/s), and was taken via video camera. Velocity is squared in the equation
and the expected sign on the coefficient is positive because as velocity increases so does kinetic
energy which has been observed to result in better vaults.
CONTACT- “Angle of contact is a measure of the body’s position when the pole contacts the
back of the plant box… Using the perpendicular line (90 degrees) from the runway as the zero
degree mark, any line drawn that tilts toward the pit is recorded as a positive angle and
conversely, if the body is angled back away from the pit it is recorded as a negative angle”
(Nielsen).
DEPARTURE- “Like the angle of contact, angle of departure uses the perpendicular line to the
runway as the zero point for reference measurement of the body angle just prior to the takeoff
foot leaving the ground” (Nielsen). The image below shows how departure angle is measured.
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In the regression CONTACT and DEPARTURE are summed together and the expected sign on
the coefficient is positive because an effective vaulter will want a large takeoff angle, and will
want to maintain this positive angle at the point of departure. This variable picks up on a vaulters
efficiency of energy transfer. If this is done correctly, the vaulter will minimize the energy lost at
departure by maintaining a tall takeoff with high potential energy.
EXTENSION- “The measurement of time is taken from the instant the athlete achieves an
inverted straight body position (hip near top hand) until the pole is fully straightened” (Nielsen).
This variable has a positive expected sign. The term “beating the pole to vertical” is often used to
describe a larger extension time, where the vaulter is able to better take advantage of the strain
energy in the pole (McGinnis). The image below shows the starting point of extension time.
INVERSION- “This is a measurement of the angular position or line of the body at the instant of
the pole straightening with the ground. A line is drawn from the top of the sternum (jugular
notch) to the crotch” (Nielsen). The expected sign on this coefficient is positive because the
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Maulin: Predicting the Pole Vault
ultimate goal for a vaulter is to be completely vertical (perpendicular to the ground) so the closer
the angle is to 90 degrees the more effective the vault is.
HIP- “The measurement is the difference in centimeters between the top of the top hand at pole
release to the maximum hip height. The hip height is measured at the greater trochanter of the
lower hip (if one is lower than the other)” (Nielsen). The expected sign on this coefficient is
positive because if the vaulters hips are higher than their grip on the pole then their probability of
clearing higher bars increases significantly because of their position in the air.
GENDER- This is a dummy variable taking the value 0 for male, and the value 1 for female.
The coefficient on this variable is expected to be negative because as can be seen women vault
lower than men, and this can be attributed to different distribution of weight and muscle strength.
Developing the Model
One of the main concerns while constructing this model is that the regression needs to
live in the world where physics applies. This issue made it so that finding the correct functional
form was bounded by these natural laws. Something that was particularly interesting was the
idea that it is possible to predict whether or not a vaulter should clear the bar given the variables
that were listed. Since the dependent variable in this equation is whether or not a vaulter misses
or makes the bar, a binary dependent variable model needed to be used. For this study the logit
model was used.
Regression Analysis
The regression that is most significant is:
SUCCESS=B0+B1(VELOCITY^2)+B2(CONTACT+DEPARTURE)+B3(EXTENSION/INVERSI
ON)+B4(INVERSION)+B5(HEIGHT)+B6(HEIGHT*GENDER)+B7(HIP)+B8(HIP*GENDER)
Table I shows the coefficients, standard errors, z-statistics, and p values for each of the
given variables in the equation listed above. All of the signs on these variables are as expected.
Because of measurement difficulties, p-values of less than .15 are considered statistically
significant for this study.
Table I
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Variable
Coefficient
Standard Error
Z-Statistic
P-Value
C
16.50686
2.940755
5.613138
0.0000
VELOCITY^2
0.090049
0.028946
3.110900
0.0019
CONTACT+DEPARTURE
0.018360
0.011180
1.642135
0.1006
EXTENSION/INVERSION
-218.9953
112.3033
-1.950034
0.0512
INVERSION
0.051532
0.009782
5.267889
0.0000
HEIGHT
5.233242
0.659646
-7.933423
0.0000
HEIGHT*GENDER
-1.013991
0.249662
-4.061459
0.0000
HIP
0.010814
0.006347
1.703917
0.0884
HIP*GENDER
0.020550
0.009999
2.055246
0.0399
The variable VELOCITY squared has a p-value of 0.0019 which is statistically
significant (p < .15). This suggests that there is a nonlinear relationship between velocity and
successful vaults. It shows that as velocity increases a vaulters chance of success increases, by an
increasing amount. The coefficient, 0.090049, is positive as expected. Because the logit model
was used for this study, dividing the coefficient by 4 will give us the marginal change of success
by a 1 unit increase or decrease of the given variable. Therefore, a 1 unit increase in the variable
VELOCITY^2 increases the chances a vaulter will clear the bar by 2.2512%. To put this into
perspective let’s take a hypothetical female vaulter who’s average velocity is 8m/s, therefore
velocity squared is 64. If this vaulter were to increase her average last 5 meter velocity before
takeoff by 0.1 so that her new velocity is 8.1m/s and her velocity squared equals 65.61. Her
velocity squared has increased by 1.61 units, and her chance of clearing the bar has increased by
3.6244%. Something important to note is that as a vaulter gains more experience it is more
difficult to increase velocity, but there is a larger payoff for increasing an already high velocity.
According to a study done by Dr. Peter McGinnis, the best male vaulters have velocities that
reach 9.5m/s and the best female vaulters have velocities that reach 8.2m/s (McGinnis). This can
help to explain why the male world record vault is much higher than the female world record.
The calculated difference in VELOCITY squared is 23.01 units. This suggests that a male with a
velocity of 9.5m/s is 51.80% more likely to clear the same bar as a female with a velocity of
8.2m/s given that all the other variables are held constant for the two individuals.
The variable CONTACT + DEPARTURE is the sum of the contact angle and the
departure angle. This variable suggests that there is a positive effect on the vault when the sum of
these variables is larger. Both contact angle and departure angle are assumed to have a positive
effect when the angle is greater, so when these are summed together they should also result in a
positive effect. This variable confirms the idea that the more the vaulter pushes the pole forward,
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Maulin: Predicting the Pole Vault
the more likely they are to clear the bar. The p-value on this variable is 0.1006 which means that
it is statistically significant (p < .15), and therefore we can conclude that it is different than zero.
For a 1 unit (1degree) change in the variable CONTACT + DEPARTURE there is a 0.459%
better chance of clearing the bar. This shows that although this variable is statistically significant,
there is not a huge impact on the vault from small changes. However, if a vaulter were to change
their CONTACT+DEPARTURE by 10 units (which could consist of increasing each angle by 5
degrees), there is a 4.59% increase in chance of clearing the bar, holding all else constant.
The individual variable INVERSION has a p-value of 0.000 which means that it is
statistically significant. The coefficient, 0.051532, is positive as was expected. This suggests that
as a vaulter increases their inversion angle there is a positive effect on chance of success.
However, this variable has a practical limit that should be noted. Any angles larger than 90
degrees would not be successful, or even possible.
The variable EXTENSION/INVERSION has a p-value of 0.0512, which is statistically
significant at the 85% confidence level. The coefficient, -218.9953, is negative which shows that
there are diminishing marginal returns on INVERSION. This means that as INVERSION
increases there is a benefit on the vaulters potential success, however this benefit increases by
less and less as INVERSION gets closer to 90 degrees. This suggests that the slope of
INVERSION is not constant. There is also a strong relationship between EXTENSION and
INVERSION that needs to be acknowledged. These two variables are highly correlated, and a
better vaulter will have a high INVERSION and a larger EXTENSION.
The next variable, HEIGHT, is also highly statistically significant with a p-value of
0.000. The coefficient, -5.233242, is negative as expected. As we increase HEIGHT by 1 unit
there is a 130.831% lower chance that a vaulter will clear the bar. This may seem extremely
high, but when we consider what 1 unit is for the variable HEIGHT this is consistent with our
expectations. A 1 unit increase in height would be 1 meter (approximately 3.28 feet). This 1 unit
increase is impractical in a real world setting because the bar is rarely raised by that much at one
time. Although a 1 unit increase is unrealistic, it is more useful to look at a 10 centimeter
increase in bar height. If we divide -130.831% by 10, this gives us the marginal effect of raising
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the bar by 10 centimeters. For every 1/10 change (10 centimeter increase), the probability that a
vaulter will clear the bar decreases by 13.0831%.
The variable HEIGHT*GENDER is capturing the fact that the slope on the variable
HEIGHT is not the same for women and men. This variable has a p-value of 0.000 which means
that it is statistically significant. The coefficient is -1.013991, and for a 1 unit increase on the
variable there is a -25.3498% effect on the probability of clearing the bar for women. If we
divide this number by 10 the result is a 2.53498% lower chance of clearing the bar for a 10
centimeter bar height increase. We then add this value to the value we got for a 1/10 change for
both genders. For every 1/10 of a unit (10 centimeter) increase in bar height, women have a
15.61808% lower chance of clearing the bar holding all else constant.
The variable HIP has a p-value of 0.0884 which means that it is statistically significant at
the 85% level (p < .15). The coefficient, .010814, is positive, which is as expected. For a 1 unit
increase in HIP there is a .4535% increase in probability of clearing the bar. This does not sound
significant at first, but the variable HIP is measured in centimeters, a small unit of measurement.
As the vaulter increases their hip height by a larger amount the effects are seen. For a 15.24
centimeter (1/2 foot) increase in HIP there is a 6.91134% better chance that a vaulter will clear
the bar.
The variable HIP*GENDER has positive coefficient of 0.020550 and a p-value of 0.0399
which means that it is also statistically significant at the 85% confidence level (p < .15). This
coefficient expresses that as a female is able to increase hip height, there is a larger increase in
probability of clearing the bar than for an increase in hip height for males. For a 1 unit increase
in HIP*GENDER there is a 0.5137% higher chance that a female will clear the given bar. This
means for a 15.24 centimeter (1/2 foot) increase, there is a 7.8288% greater probability of
success for a female. If we sum the benefit from the variable HIP with the variable
HIP*GENDER we get that there is a 14.740128% increased chance of success for 15 centimeter
increase in hip height for females. This variable captures the fact that females have different
body types than males, and given this different center of gravity, hip height affects the vault
differently. It is important to note that males often have more upper body strength than females,
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Maulin: Predicting the Pole Vault
so it is physically easier for a male to increase HIP. Male values for HIP tend to be significantly
larger.
Forecasting
In order to see how the model does when forecasting whether or not a vaulter is going to
clear the bar, 20 observations were taken out of the sample and forecasted in a regression
analysis program. A new variable SUCCESSF was created. This variable SUCCESSF included
the 20 observations that were taken out of the regression to begin with in order to see how well
the model predicted these values. This forecast guessed only 3 values incorrectly for the 20
observations. That means that for this specific forecast, it was correct 85% of the time. (See
Table II).
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Table II
SUCCESSF
0.09557
0.609155
0.687094
0.408421
0.286314
0.378055
0.564304
0.651476
0.569849
0.309585
0.263761
0.524475
0.685238
0.459827
0.497464
0.410221
0.579463
0.607856
0.437247
0.540094
SUCCESSF < .5 = 0 AND
SUCCESSF > .5 = 1
0
1
1
0
0
0
1
1
1
0
0
1
1
0
0
0
1
1
0
1
SUCCESS actual
0
1
1
0
0
0
1
1
1
0
0
0
1
0
0
0
1
1
1
0
Overall, however, this model correctly predicted 356 values out of 522 observations. This
means that 68.1992% of the time the model correctly predicts whether a vaulter is going to miss
or make the bar. Graph I is a representation of the predicted values for vaults that we know were
makes. Graph II shows the predicted values for vaults that were unsuccessful. The forecasted
values are between 1 and 0. If the predicted value is less than 0.5 then it takes the value 0, and if
the predicted value is greater than 0.5 it takes the value 1.
Graph I
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Maulin: Predicting the Pole Vault
Values above the line at 0.5 were correct predictions.
Graph II
Values below the line at 0.5 were correct predictions.
Graph I and II illustrate the point that the model is much better at predicting when a
vaulter is going to miss the bar than if they will make the bar. This means that the model can
predict better when a vaulter is doing something incorrectly and should not make the bar. Often
other factors besides pole vault technique lead to a vault not being successful. One of the major
issues is that there was no data gathered on the athletes’ standards setting which has led to a less
accurate model. In the pole vault, the athlete has the ability to choose how close the bar is to the
front of the pit based on how the standards are positioned. This has the potential to make or
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break a vault depending on where the bar is located in space. These settings are limited to a
range of 45cm to 80cm. A vaulter could be doing everything right according to this model and
still miss the bar if the standards weren’t set in the correct location. Another factor that could
lead to a prediction being incorrect is if the vaulter hits the bar with some part of their body on
their exit off of the pole. It is easy for the athlete to brush the bar with their arm or another part
of their body. Also, a vaulter may do everything correct, fail to throw their pole, and have the
pole fall back and hit the bar.
Practical Applications
This model has the potential to be used as a coaching tool. If a vaulter videos an
individual vault, and calculates the values for each of these variables they can be plugged into
this equation. Holding all variables constant, a vaulter can change one value, and see what the
increased probability is for success. Workout programs can then be structured around which
values are most easily improved, and focus can be placed on the most important variables for
success. This equation suggests that velocity has the potential to significantly increase success in
the vault. For beginning vaulters, velocity can be increased with repetition. As a vaulter gets
more experience this equation helps to direct what fine tuning will produce the most success.
Conclusion
This is an area of Track and Field that could use even further study. Mounting a camera
at the same distance from the same vault pit would help to reduce measurement errors. This
could help to lessen the impacts of measurement errors because these vaults will be able to be
compared side by side from the same camera location. Adding a new variable for standards will
increase predicting capabilities; however, this model will never be able to predict close to 100%
of the time because of the sheer randomness that is the vault. Other variables that should be
included in further studies are pole grip height, pole size, vaulter body weight and height, and
wind speed (for outdoor meets).
Acknowledgements
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Maulin: Predicting the Pole Vault
The data for this study was provided by Dave Nielsen. Dave Nielsen is currently the Head Coach
of the Idaho State University Track and Field team, and was an excellent pole vaulter in college,
as well as coach to many notable pole vaulters including the first women’s Olympic gold
medalist, Stacy Dragila. He has done countless studies on the mechanics of pole vault, and his
work has led to many important changes in the pole vault regarding safety concerns and proper
equipment usage.
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Maulin: Predicting the Pole Vault
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