THE BICYCLE RACE ALBERT SCHUELLER. INTRODUCTION We will conider the ituation of a cyclit paing a refrehent tation in a bicycle race and the relative poition of the cyclit and her chaing upport car. The cyclit ride at a contant velocity with her ditance in eter fro the refrehent tation given by where i tie in econd after paing the tation. The puruing upport car ditance in eter fro the tation i given by. There are everal apect of the ituation we will analyze. We are given the poition of the cyclit relative to the location of the refrehent tation a a function of tie. We denote by the poition of the cyclit at tie. Denote by the poition function of the auto. Fro the dicuion in. of Calculu, Stewart rd ed, we know that the velocity of the cyclit i and the velocity of the auto i. In of thi article we copute the cyclit velocity and the tie at which the chae car catche the cyclit. In we explore the effect of different cyclit velocitie on eeting tie. In we attept to illutrate the variou cenario graphically. Finally, in we connect a theortical reult about cubic polynoial with the ituation of the cyclit and the chae car.. VELOCITY AND MEETING TIME In thi ection we deterine the peed of the cyclit and how long it take the chae car to catch her. We have the additional inforation that when the chae car catche the cyclit their velocitie are the ae. Thi provide u with two algebraic equation. When the chae car catche the cyclit their poition are the ae, thu Furtherore, when their poition coincide, by deign, their velocitie are the ae, thu
ALBERT SCHUELLER Together, we have two equation in the two unknown and. () ()! " Uing the equation (), we eliinate fro the equation () and olve for. We find two olution, and $#. The firt olution i conitent with the fact that their poition coincided at the refrehent tation. The econd tie, econd, i when the upport car catche the cyclit. The velocity of the cyclit a a function of tie i given by %, where i a contant. Deterining will give the velocity of the cyclit. We know that at &# equation () hold. We can deterine by ubtituting '# #)( into equation ().#)( Doing o we find /. Thu the cyclit i traveling at a rate of /.. DIFFERENT VELOCITIES We will now conider what happen if i allowed to vary. We will derive a forula for in ter of for the tie when the poition of the cyclit and the car coincide. By etting the two poition function equal to each other and olving for, we can deterine the length of tie it take for the puruing car to catch the cyclit. Fro thi calculation we ee that * +, - /. - /. i one olution, which i conitent with the fact that their poition are the ae when the cyclit pae the refrehent tation. We ee that there are potentially two ore tie where the car and the cyclit have the ae poition. To deterine thee tie, we ut apply the quadratic forula, :9; 9 -
where 9, and ; THE BICYCLE RACE. Doing o we have ) In order for the olution to be real valued, we ut have +<. Auing >=, we ee that the expreion for i alway poitive and yield two #)( ditinct tie. Fro thi we ay conclude that if he ride fater than #( /, the upport car will not catch her. If he ride lower than /, the upport car will eet her twice. #)(. GRAPHICAL ILLUSTRATION In thi ection we provide graph to illutrate what happen in the firt cae where the car and the cyclit eet and have the ae velocity and in the econd cae where the velocity ay change. Figure, how#)( the poition function of the cyclit and the car on the ae axe with / the optial peed. We ee that after the two function interect. only once at?# FIGURE. A@ Figure ( how the ae graph with the cyclit velocity lightly fater at B /. In thi cae we ee that the two plot fail to interect after. Figure ( how the ae graph with the cyclit velocity lightly lower at /. In thi cae we ee that the two plot interect twice after.
ALBERT SCHUELLER FIGURE. AC FIGURE. AD In the ituation depicted in Figure, it would be difficult for the upport car to give refrehent ince there would till be oe ditance between the car the cyclit. In the ituation depicted in Figure, it would be difficult for the car to give refrehent becaue the velocitie of the cyclit and the car would be too different.. CUBIC POLYNOMIALS We will now how that if a cubic polynoial, E root at 9, then EF 9 G relationhip between the cyclit and the auto. Let 9, ha a repeated real. We will then ue thi reult to interpret the be a double root. If
I i the reaining root of E E HJI K9 I where i the coefficient on the E HJI "L9 Hence, EM 9 H. THE BICYCLE RACE, then we ay write KL ter. Thu, "L9 KL Thi reult i relevant to the firt ituation in which the car and the cyclit are traveling at the ae peed when they eet. The cubic polynoial in quetion i E N O and repreent the difference in the poition #)( of the cyclit and the car. Uing the factor() call in aple with / we have E H P #) We ee that Q# i a double root of E. Notice further that ER give the difference in the velocitie #Sof the cyclit and the auto. The above double root reult tate that E which i conitent with the fact that at the tie that they eet the cyclit and the auto are traveling at the ae peed. In the later exaple where i allowed to vary, we find that E ha not double root and the delicate condition of atching poition and peed i detroyed.