Go for the Gold!

Date: 1/28/96
From: Anonymous
Subject: Math Problem to Solve

Hi Dr. Math.  My math problem is a long one so here goes.  It is a 
word problem titled Go For the Gold.
      You are one of the women's track coaches for the United States
Olympic team.  Your job is to prepare the athletes for the 1,500-meter
race.  You decide to analyze races from past Olympic games to better 
inform yourself of race strategy, runners' speeds, and athletic 
conditioning.  You obtain a video tape of the women's 1,500-meter race 
from the 1992 Olympic games.  After you and your assistant view the 
video tape, she makes a table of the race data for you to use.  This 
table is shown in the Data Bank.

Your task is to analyze the race.  Focus in on at least five of the
runners.  Determine how far the runners run in a period of time.  Use 
that data to create a graph or series of graphs that show their speeds 
over time.  On the graphs, graph distance v.time and distance v. 
speed. Your graphs may focus on more than one runner.  Illustrate 
intervals of acceleration, deceleration, and speed.  Note where the 
runners overtake other runners.  Calculate the average speed and rates 
of acceleration of the runners.
        Now, here's the 1992 Women's 1,500-Meter Data DATA BANK

Order     Name             Country     Time     Time    Time     Time
                                      at 400m  at800m  at1200m at1500m

1   Boulmerka, Hassiba    Algeria   1:00.65  2:05.61  3:09.92  3:55.30

2   Rogacheva, Lyudmila  Unified Tm.1:00.55  2:05.02  3:09.88  3:56.91

3   Qu, Yunxia            China     1:01.34  2:06.13  3:10.41  3:57.08

4   Dorovskikh, Tatiana  Unified Tm.1:01.95  2:06.87  3:11.24  3:57.92

5   Liu, Li               China     1:03.64  2:08.91  3:13.76  4:00.20

6   Zuniga, Maite         Spain     1:04.36  2:09.79  3:14.06  4:00.59

7   Rudz, Malgorzata      Poland    1:05.97  2:10.69  3:15.49  4:01.91

8   Podkopaye,Yekaterina Unified Tm.1:04.26  2:10.48  3:16.01  4:02.03

9   Mutola, Maria de L.  Mozambique 1:06.23  2:12.78  3:16.99  4:02.60

10  Plumer, Patti Sue    USA        1:06.19  2:12.59  3:16.42  4:03.42

11  Fidatov, Elena       Romania    1:07.59  2:13.29  3:17.82  4:06.44

(Remember these formulas when solving:  d=rt   and   s=d/T)

Thank you.  I appreciate your help in solving this problem.  This is 
an investigation.  Good luck.

Date: 9/28/96
From: Doctor Sydney
Subject: Re: Math Problem to Solve

Hello!  This is a great question .  The problem asks you to focus in 
on at least five runners, but for now I will just talk about one 
runner; you can use the same techniques to analyze the data on the 
other runners.  Let's analyze the data on Elena Fidatov of Romania 
(#11).  

How far did she run in each period of time?   Well, this is hard to 
answer precisely, given the data above.  For instance, if we want to 
know how far she ran in the first minute, we have to estimate because 
our first piece of data says that she ran 400m in 1:07.59 minutes.  
So, assuming that she ran at a constant speed (an assumption that is 
probably false, but is the most accurate assumption we can make), 
let's figure out her average speed in the first 1:07.59 minutes of 
running.  Well, we know that in 1:07.59 minutes she ran 400 m, right?  
And, speed = distance/time, so her average speed for the first 400 m 
is 400m/67.59 seconds = 5.9 m/s.  To figure out how far she ran in 1 
minute, then, we again use the formula speed = distance/time; this 
time we know the speed and the time but we don't know the distance, so 
our equation becomes:  

5.9 m/min = distance/60 s  

So, the distance covered in 1 minute is about 354 m.  

Now, using these techniques, you can calculate a guess for how far 
Elena ran in any given amount of time by assuming her speed has been 
constant from the start of the race.  Once you have calculated how far 
she has run for lots of different times, you can plot the values on a 
graph with the time on the x-axis and the distance on the y-axis.  If 
you need help on plotting graphs, write back.  For now, I'll assume 
you know how to do this.

You should notice that as you were calculating how far Elena ran in 1 
minute, you also calculated her average speed over the course of 1 
minute, right?  So, if you keep track of the average speed at any 
given point in time and also the distance covered at that point in 
time, you can create another graph in which distance is on the x-axis 
and speed is on the y-axis.

Now look at your graphs.  From your graphs you can figure out when 
Elena is speeding up and when she is slowing down?  Can you tell how?  
I'll let you think about this one.

By doing the above process for a number of runners, you can compare 
the different graphs to see how is overtaking who, when.  I have 
answered most of your questions.  Keep thinking on the other ones!  
Once you have the graphs in front of you, things should be clearer!  
Write back if you have more questions.

-Doctor Sydney,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/