The Bicycle Race

THE BICYCLE RACE 1. I NTRODUCTION We will consider the situation of a cyclist passing a refreshment station in a bicycle race and the relative positions of the cyclist and her chasing support car. The cyclist rides at a constant velocity with her distance in 

meters from the refreshment station given by where is time in seconds after passing the station. The pursuing support car’s distance in meters from 

      the station is given by  . There are several aspects of the situation we will analyze. We are given the position of the cyclist relative to the location of the re
     freshment station as a
function of time. We denote by the position
 of the cyclist at time . Denote by   the position function of the auto. From the discussion in  2.3
of Calculus, Stewart 3rd ed, we know
 that the velocity of the cyclist is   and the velocity of the auto is   . In  2 of this article we compute the cyclist’s velocity and the time at which the chase car catches the cyclist. In  3 we explore the effect of different cyclist velocities on meeting times. In  4 we attempt to illustrate the various scenarios graphically. Finally, in  5 we connect a theortical result about cubic polynomials with the situation of the cyclist and the chase car. 2. V ELOCITY AND MEETING TIME In this section we determine the speed of the cyclist and how long it takes the chase car to catch her. We have the additional information that when the chase car catches the cyclist their velocities are the same. This provides us with two algebraic equations. When the chase car catches the cyclist their positions are the same, thus

      Furthermore, when their positions coincide, by design, their velocities are the same, thus

       1 2 Together, we have two equations in the two unknowns (1) (2)  

      
" 
  ! 

and . Using the equation (2), we from # the equation (1) and solve for

 eliminate
$  s and . We find two solutions, s. The first solution is consistent with the fact that their positions coincided at the refreshment station. The second time, 5 seconds, is when the support car catches the cyclist.
% The velocity of the cyclist as a function of time is given by    , where is a constant. Determining will give the velocity of the cyclist.
&# We know that at s equation (2) holds. We can determine by substi #)( 
'# tuting s into equation (2). #)Doing so we find m/s. Thus the (  cyclist is traveling at a rate of m/s. 3. D IFFERENT VELOCITIES We will now consider what happens if is allowed to vary. We will
derive a formula for in terms of for the times when the position of the cyclist and the car coincide.
By setting the two position functions equal to each other and solving for , we can determine the length of time it takes for the pursuing car to catch the cyclist.
*
       

+
    

  
  
,
-
/ .     -
/ .

10     
  is one solution, which is consisFrom this calculation we see that tent with the fact that their positions are the same when the cyclist passes the refreshment station. We see that there are potentially two more times where the car and the cyclist have the same position. To determine these times, we must apply the quadratic formula, 32465 2 87:9 ;
 9 THE BICYCLE RACE where 9  , 2  ;   and
 3  . Doing so we have 4 5   )   #)(  +< In order>= for the solutions to be real–valued, we . As
must have  suming , we see that the expression for is always positive and yields two distinct times. From this we may conclude that if she rides faster than #)(  #(  m/s, the support car will not catch her. If she rides slower than m/s, the support car will meet her twice. 4. G RAPHICAL ILLUSTRATION In this section we provide graphs to illustrate what happens in the first case where the car and the cyclist meet and have the same velocity and in the second case where the velocity may change. Figure 1, shows  #)the (  position functions of the cyclist and the car
on the same axes with   s the two functions intersect m/s–the optimal speed. We see that after ?  #
s. only once at 60 50 40 m 30 20 10 0 1 2 F IGURE 1.  3 4 s 5 6 7 8 A@  , Cyclist–Solid Line, Auto–Dotted Line. Figure  ( 2 shows the same graph with the cyclist’s velocity slightly faster B at m/s. In this case we see that the two plots fail to intersect after
  s. at
 Figure 3( shows the same graph with the cyclist’s velocity slightly slower    m/s. In this case we see that the two plots intersect twice after  s. 4 70 60 50 40 m 30 20 10 0 1 2  3 4 s 5 6 7 8 AC  , Cyclist–Solid Line, Auto–Dotted Line. F IGURE 2. 50 40 30 m 20 10 0 1 2 F IGURE 3.  3 4 s 5 6 7 8 AD  , Cyclist–Solid Line, Auto–Dotted Line. In the situation depicted in Figure 2, it would be difficult for the support car to give refreshment since there would still be some distance between the car the cyclist. In the situation depicted in Figure 3, it would be difficult for the car to give refreshment because the velocities of the cyclist and the car would be too different. 5. C UBIC POLYNOMIALS
  , has a repeated real We will show that
 now 9 9 G if a cubic polynomial, E  . We will then use this root at , then EF  to interpret the
 result 9 relationship between the cyclist and the auto. Let be a double root. If THE BICYCLE RACE 5
 is the remaining root of E  , then we may write
HJI
K89 
KL2    E  I
 where is the coefficient on the term. Thus,
HJI
"L9  0 I
"L9 
KL2     E  9 H  . Hence, EM  This result is relevant to the first situation in which the car and the cyclist are traveling at the speed when 
Nsame

 they meet. The cubic polynomial O     and represents the difference in the in question is E  positions  #)(  of the cyclist and the car. Using the factor() call in maple with m/s we have
H 

P #)  E   
Q# We see that is a double root of E . Notice further that ER gives the difference in the velocities #Sof the cyclist and the auto. The above double  which is consistent with the fact that at the root result states that E  time that they meet the cyclist and the auto are traveling at the same speed. In the later examples where is allowed to vary, we find that E has not double roots and the delicate condition of matching position and speed is destroyed. 2