We wanted to model the Long Jump track & field event to see if we could mathematically incorporate technical elements to the simple projectile motion model and see tangible differences in jump distances.\u00a0 We also wanted to model the effects of air, since we thought that wind speed and air density must have some effect on jump distances, but did not know how to quantify its impact.\u00a0 We believed we could\u00a0achieve these goals with a comprehensive Matlab program, and Chapter 2\u00a0of Giordano and Nakanishi\u2019s Computational Physics<\/i> on realistic projectile motion gave us a clear starting point.<\/p>\n
Program Description<\/strong><\/p>\n This program simulates the long jump track and field event to observe\u00a0changes in jump distance as related variables are adjusted. The variables we were\u00a0most interested in are: air density, to determine if jumps done at\u00a0different altitudes are comparable; wind speed, to observe how much wind\u00a0resistance can affect performance; whether or not the\u00a0athlete cycles his or her arms in the air, to see how this movement\u00a0affects body position in the air; and, of course, the final jump distance. \u00a0We could also adjust the size of our athlete using our program, but\u00a0the adjustments, as long as they are\u00a0kept within reasonable limits, would have a negligible effect on the results. \u00a0The athlete\u2019s physical proportions are based off of our very own Kadeem Nibbs, who always wanted to participate in the long jump but never could\u00a0due to scheduling conflicts.<\/p>\n We originally wanted to use 3D models to run all of our tests, but working with 3D shapes and deforming them proved difficult. \u00a0We decided to use point particles for the tests where they provide an accurate approximation (wind assistance and air density), and then 2D \u201cpatch\u201d shapes for tests where body position became exceedingly important (limb cycling). \u00a0For the trial where the athlete did not cycle his limbs, we created one fixed-shape rectangle to model the athlete\u2019s body, as if he were held rigid throughout the entire jump. We modeled the athlete with two rectangles when limb cycling was allowed, one rectangle to represent the torso, and another for the legs, so that they could move independently.<\/p>\n Final Results<\/strong><\/p>\n Real World properties and their affect on Long Jumping:<\/strong><\/p>\n While our preliminary<\/em> results showed that wind had a major impact on the final jump distance, with a difference of 7m\/s resulting in a change of approximately 2 meters in jump distance, we found that this was due to a missing square root sign when calculating velocity. \u00a0When this was fixed, we found that the same difference in wind speed accounted for a difference in inches<\/em> in jump distance.<\/p>\n Our Resulting Projectile Motion with Varying Wind Speed:<\/b><\/em> Our Resulting Projectile Motion with Varying Wind Speed (close up view):<\/b><\/em> A 0.1 change in meters is about a difference of 4 inches. While this may seem negligible on a macroscopic level, the top two World Records in the long jump only differ by 4.5 inches. \u00a0So a fortuitous wind gust may be the difference between a gold medal and nothing.<\/p>\n For our second real world property we found that air density had a negligible effect on the jump distance, as\u00a0variations of up to 50% in air density resulted in less than a millimeter of difference. This reaffirms what was\u00a0learned in introductory mechanics: that air resistance is negligible in most cases of projectile motion, with exceptions being when the air is moving and when the object is moving at high speeds.<\/p>\n Our Resulting Projectile Motion with Varying Air Density:<\/b><\/em> Air resistance will not significantly affect an athlete jumping at 20mph into still air. This also shows that although air density can be as high as 1.2kg\/m^-3 at cities near sea level, and as low at .75kg\/m^-3 at cities 5000m above sea level, long jumps performed at any city in the world can be compared because of air density\u2019s negligible effects on performance.<\/p>\n Modeling the Human with the Patch Mechanism:<\/strong><\/p>\n After thoroughly researching track and field coaching manuals and websites, we learned that the purpose of cycling one\u2019s arms and legs while in the air is to keep one\u2019s torso from rotating forward. The torque generated during an athlete\u2019s takeoff typically generates a large forward angular momentum. As a result, if an athlete\u00a0does not cycle their arms\/legs properly while midair, they may end up tilting too far forward, hitting the ground face first, and losing some of their teeth. This is demonstrated in the figure below\u00a0when our code is run.<\/p>\n
\n![\"Clip1better\"](\"http:\/\/pages.vassar.edu\/magnes\/files\/2015\/05\/Clip1better.gif\")
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