Tour de TNB

Figures 1 & 2: Sharp Curves Along the Tour de France. (Photo Credits: AP wire and Graham Watson).

1. Instructions

This lab is tentatively due Thursday, June 16th in recitation.  TA's will have lab office hours that week (June 13th - 16th).  The TAs in the lab are there to help you with Mathematica syntax; they are not there to debug your files.  Please keep this in mind.

Late labs are not accepted. You are encouraged to work in groups of no more than 3 people, though you can work alone.  You need hand in only one lab per group.  One electronic copy and one hard copy must be submitted.

On your title page, clearly mark

Format is worth 20% of your grade.  Please refer to the following writing guidelines for the expository sections of this report, You are required to know and follow the Writing Guidelines for all labs.

Writing Guidelines for Calculus Labs
Please refer to the "Projects" tab on the course webpage for sample write-ups.

PLEASE DO NOT LEAVE ANY OF YOUR FILES SAVED ON THE COMPUTERS IN THE LAB!
Either e-mail them to yourself or save on a USB 'memory stick'

If you have never used Mathematica, you may want to have a look at these tutorial files - they will help with this lab project.  In addition, you may want to look at the information on the web page for APPM 2450 (The optional lab course).  You will find samples of all the commands and syntax you will need in the notebooks below or on 2450's web page.

2. Background

In this lab you will help improve the safety of a proposed Tour de France stage. Every year, nearly 200 of the world's premier cyclists gather to participate in this landmark road race across France. The Tour de France takes place over three weeks in July and the daily stages change from year to year. Technological improvements have contributed to significantlyfaster average speeds and so rider safety has increasingly become a focus in selecting stages for the next year's race.

2a. Sharpest Curves

The sharp curves along a route are of particular concern to race organizers. Tight corners, switchbacks and traffic circles have become fixtures of the race and are especially popular with spectators for the difficulty they present speeding cyclists. Despite (and/or contributing to) their popularity, these curves quickly become treacherous after a rainfall or, more often than not, when the tar of the asphalt begins to melt in the heat of July. Given that crashes along these stretches will occur, organizers would like to minimize the time needed for a fallen rider to recover and rejoin the race. An effective way to do this is to place a set of hay bales along the sides of the sharper curves. However, because of the cost of these measures, organizers can only provide one full set of hay bales per race hour. Consequently, organizers have decided to place the bales on the "most dangerous" (in some sense) curve encountered by the racers in a given hour to avoid further damage to riders and their bikes in scenes such as that in Figure 3.
Figure 3: A rider succumbs to a sharp curve. (Photo Credit: Stuart Peplow, Westerley Cycling Club).

2b. Feed Zones

It is often the case that, even after three weeks of racing, only a few seconds separate the first place and second place riders at the end of the Tour de France. To meet the racers' nutritional demands, while still preventing unnecessary delays, a "feed zone" is placed on each stage. The feed zone is a small stretch of roadway where racers are handed bags of energy-laden food which they consume on the go (See Figure 4). The feed zone can often be a dangerous stage of the race as riders continue to jockey for position while reaching over to grab a bag. To make matters worse, the section of the course after the feed zone is often lined with local children hoping to catch a souvenir as riders discard their bags. Organizers are hence forced to place the feed zone for a particular stage in the "straightest" portion of the race.
Figure 4: Greg LeMond enjoying lunch on the 1992 Tour. (Photo Credit: Robert Pratta, Reuters).

3. Problem Statement

For the 2010 Tour de France, race organizers are considering adding a race stage which includes an hour-long stretch between the villages of Chateau de Cauchy and Laplace. In this lab, you will help organizers determine where to place both a feed zone and their single allotment of hay bales by analyzing the proposed route using vector Calculus. You have been told that neither of these items should be placed within 1 km of either village since they would interfere with the spectators there. If changes in elevation are neglected, the proposed route can be (approximately) described by the following position vector (in km):
r(t)=3.5 Cos[3.6 Pi t] i + (Cos[4 Pi t] + 5 t) j,
where the race is projected to pass through the center of Chateau de Cauchy at time t=0 hours and arrive in Laplace at time t=1 hours. All answers should include units and should be in terms of kilometers (km) and hours (h).

Before making your recommendations, you should answer the following questions:

  1. One of the more interesting features of this route is that riders cross the Pont de Pascal which actually passes over a section of the route which they later race on. Plot the route and label the two villages and the Pont (Hint: Pont = Bridge) on your plot. Briefly state your opinions on the validity of the position vector as an approximation (ie.- do you think the position vector is realistic? Why or why not?).
  2. Plot the (projected) speed of the riders along the course as a function of time. What is the average speed along this stretch of the route?  (Hint use NIntegrate here!  In Mathematica, type
    ?NIntegrate for mor    e info)
  3. Determine both the length of the route that racers take and the direct distance between the two villages ("as the crow flies").
  4. Determine and plot the curvature (kappa) of the proposed route (as a function of time). What do you notice about the relationship between this plot and the plot of speed? Does this make sense?
  5. Determine and plot the normal component of acceleration of riders on the proposed route (as a function of time). Comment on the relationship of this plot with the previous plots.
  6. Recalling that organizers do not want to place either of the two components too close to the two villages, determine at what times the riders are projected to be 1 km away (along the route) from each of the villages. (Hint: Mathematica may not like an integral that you want to use here, you may just have to get these numbers via "brute force"...)
  7. Determine a coordinate in space and the time when riders are projected to reach the point where maximum normal acceleration occurs. Explain (briefly) the logic behind placing the hay bales at this point. (No serious root finding is necessary here, a plot to a sufficient precision will suffice).
  8. Next, using the plot of curvature (ignore the normal acceleration for this question), propose and justify a location (give the coordinates, as well as time when the riders are projected to reach the feed zone) of the feed zone. Remember that riders will need to pick up, eat and discard their bag over this stretch, so you may want to give a three minute interval of time (on both sides) for this zone (Note: this could change where you put the feed zone!)

 

Hints: 

Lab Created by Stefan Wild.  Last Modified by Andy Guinn, June 2010.