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| Fig. 1: A pole vaulter using the pole to project his body over a cross bar. (Source: Wikimedia Commons) |
The pole vault is an event in track and field that involves an athlete using a long, bendable pole to vault himself over a cross bar at predetermined heights. The athlete first holds the pole on one end in his hands, and then begins his run-up. For elite athletes, this run-up is usually between 16 and 20 steps long. Once the athlete reaches the end of his or her run-up, the athlete will push the pole above their head as the end of the pole makes contact with a small metal box embedded in the ground. This impact converts the kinetic energy of the athlete into elastic energy stored in the pole. The pole then releases this stored elastic energy as it straightens out, essentially flinging the athlete into the air over the bar and onto a soft mat below (see Fig. 1). According to Nicholas Linthorne and Gemma Weetman of The Journal of Sports Science and Medicine, at its most basic the pole vault is "essentially about generating kinetic energy in the run-up and then using a long pole to convert this energy into gravitational potential energy". [1]
Using this simplified model, a quick and easy equation can therefore be used to estimate the height a vaulter will reach given his or her running velocity and center of mass height: [2]
| h | = | v2 2g |
In this equation, h = change in height of the center of mass, v = velocity at takeoff of the pole vaulter, and g = gravitational constant of 9.81m/s2. Using this equation, a number of assumptions must be made. First, we assume the bending and unbending of the pole are perfectly elastic, i.e. involve no loss of energy. Second, we assume that the vaulter is simply a point mass that does not move or contort their body. In real pole vaulting, the vaulter will jump into the takeoff and subsequently swing their body upside down in order to further load the pole and increase the stored elastic energy. This also gives the athlete momentum that allows them to vault themselves even higher over the bar.
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| Fig. 1: An illustration of a pole vaulter in the pike position. In this illustration, the crossbar is represented by the blue circle, while the vaulter's center of mass is represented by the red circle. (Source: H. Williams) |
To show an example of the equation in use, we will use Renaud Lavillenie, the current world record holder in the pole vault. Given that elite pole vaulters can reach speeds in excess of 9.5 m/s during the last steps of their approach, we can very loosely estimate Lavillenie's velocity at approach as 10 m/s. [3] Given this data and a gravitational constant of 9.81 m/sec2, we can calculate an estimated change in height of his center of mass of approximately 5.10 m. Lavillenie started his jump with his body upright, therefore we can estimate that his center of mass was roughly 1 m above the ground at the moment of his take-off, so 1 m must be added to the change in height for a total height of 6.10 m. Renaud Lavillenie's personal best is 6.16 m, showing that this equation can be used to quickly and roughly estimate pole vault height.
Given that this equation is very simplified, there are a number of factors that inhibit the equation from giving a completely accurate estimation of the pole vault. First, we are assuming the pole bends and unbends without losing energy, which in the real world would not happen. Second, at the peak of the vault the athlete pikes his or her body such that the athletes center of mass will pass under the bar (see Fig. 2). This could be a potential reason for why our equation predicted Lavillenie's ideal vault height slightly lower than his actual. His center of mass might have passed under the bar only reaching 6.10 m, but his body could have cleared the bar set at 6.16 m.
© Harrison Williams. The author warrants that the work is the author's own and that Stanford University provided no input other than typesetting and referencing guidelines. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.
[1] N.P. Linthorne and A. H. Gemma Weetman, "Effects of Run-Up Velocity on Performance, Kinematics, and Energy Exchanges in The Pole Vault," J. Sports Sci. Med. 11, 245 (2012).
[2] R. Knight, Physics for Scientists and Engineers: A Strategic Approach with Modern Physics, 3rd Ed. (Pearson, 2012).
[3] P. McGinnis, "Mechanics Of The Pole Vault," SUNY College at Cortland, 13 Dec 07.
