Lab 1_TNB - APPM 2350 EXAM 1 FALL 2010 INSTRUCTIONS:...

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APPM 2350 Calculus III LAB 1: Tour de TNB Aaron Willoughby 830-03-9918 Professor: Adam Norris T/A: Snyder 10am 022
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2 I. Introduction In order to gain a better understanding of vector calculus and its application to the world of engineering we will take a look at how specific course design elements for the 2006 Tour de France can be modeled and studied with vectors and vector analysis. We will first examine safety elements associated with sharp curves in course design. Areas with tight turns are a staple to the race and with spectators alike and thus present a cause for concern. Knowing that crashes will occur eventually at these locations, organizers would like to effectively minimize the turnaround time of fallen riders by laying the sharpest and most dangerous curves for every race hour with hay bales. We will then examine safety elements associated with a rider’s need for nutrition during the race. Organizers would like to place a “feed zone,” a place for riders to grab food while continuing to race, on the straightest portion of each stage. By modeling and manipulating race position and velocity with vectors and vector calculus, we can determine where to implement these design constituents. II. Analysis The model we will use to describe the position of this particular stage of the race from the village of Chateau de Cauchy to the village of Laplace is given below in Equation 1. Equation 1: Equation 1 can be plotted parametrically to visualize the route or path taken along the stage. The vector equation (Equation 1) appears to be an adequate approximation clearly showing straight and curved sections. The Pont de Pascal (the bridge) is located where the graph crosses itself indicating that it passes over a later section of the course. This can be viewed in Figure 1 below.
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2 Figure 1 The speed of the riders can be found by taking the derivative of the position vector to find the velocity vector and finding its vector magnitude. This is given below in Equation 2. The calculations can be found in Appendix A-1. Equation 2: To visualize the speed of the riders, we plot with respect to time. This is shown in Figure 2 below.
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2 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 T i m e n h r s n 1 0 2 0 3 0 4 0 5 0 D i s t a n c e n k m n s p e e d Figure 2 The average speed can be determined by integrating the speed with respect to time to find the total distance travelled (arc length) and dividing by the total elapsed time. This value is determined to be 36.89 ( 22.93 ); this seems to be a reasonable value for professional cyclists. The calculation can be found in Appendix A-2. The length or total distance of the stage can be found by integrating speed with respect to
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This note was uploaded on 02/27/2008 for the course APPM 2350 taught by Professor Adamnorris during the Fall '07 term at Colorado.

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Lab 1_TNB - APPM 2350 EXAM 1 FALL 2010 INSTRUCTIONS:...

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