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APPM 2350
Calculus III
LAB 1:
Tour de TNB
Aaron Willoughby
830039918
Professor:
Adam Norris
T/A:
Snyder 10am 022
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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.
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 A1.
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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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 A2.
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.
 Fall '07
 ADAMNORRIS

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