OFFWALL BRIEF ABSTRACT We attempt to determine just how we should launch a rubber ball so that it bounces of the sides of a deep vertical box and finally goes through a slit near the bottom of the box in order to win a prize. We use our laws of falling bodies many times. GENERAL INFORMATION FileName: OFFWALL Full title: Off the Wall - Just a Game Developers: Susan Clements, Terre Haute South Vigo High School. 3737 S. 7th Street, Terre Haute IN 47802 USA. Ken Kerr, Glenbrook South High School, 4000 W. Lake Ave., Glenview IL 60025 USA. Brian J. Winkel, Department of Mathematics, Rose-Hulman Institute of Technology, Terre Haute IN 47803 USA. Contact: Brian J. Winkel, Department of Mathematics, Rose-Hulman Institute of Technology, Terre Haute IN 47803 USA. Phone: 812-877-8412. Email: winkel@rose-hulman.edu. FAX: 812-877-3198. Support: The production of this material is supported by the National Science Foundation under Division of Undergraduate Education grant DUE-9352849: Development Site for Complex, Technology-Based Problems in Calculus with Applications in Science and Engineering and the Arvin Foundation of Columbus IN. STATEMENT OF PROBLEM You are at an amusement park near your town and there is this game in which participants try to get a small rubber ball into a rectangular hole at the bottom of a square-shaped, plexiglass box. The box is 40 cm on each side and 200 cm deep. The hole at the bottom of the box is on the far side and is 5 cm deep, running the length of the box, i.e.the hole is 40 cm long. From the top of the box on the near side (opposite the far side! and at a point right in the middle of the nearside edge) contestants are permitted to use a spring loaded gun to fire a small (.2 cm radius) rubber ball. The gun is fixed perpendicular to the nearside edge and parallel to the bottom of the box, i.e. horizontal. The ball is fired from the gun. The ball proceeds to bounce off the far side, then the near side, then the far side, etc. until it reaches the bottom of the box. The only variable which contestants are permitted to vary is the initial velocity (via the spring pull back mechanism in the gun) with which they shoot the ball. The gun itself is restricted to imparting initial velocities only in the range of 100 to 500 cm/sec and the calibrations on the drawback mechanisms permit the shooter to set these to the nearest cm/sec. Rarely will a contestant win and they always seem to do it by luck. You wish to study the game and do it by modeling the phenomena and replicating this in real life to consistently win. You capture images of previous contestants' efforts on your minicam and, after studying the tapes at home, you notice the horizontal velocity after it hits any wall is always .7 of what it was before hitting that wall. (This number, namely the fraction of the vertical component of the velocity an object possesses coming off the wall over the vertical component of the velocity going into the wall is called the coefficient of restitution.) Question 1: Build a model of this phenomena, being sure to state your assumptions, and show that you can win every time by prescribing a suitable set of velocities (between 100 and 500 cm/sec remember) at which you can fire the ball and land in the hole every time!! Question 2: If the typical user just selects a setting at random between 100 cm/sec and 500 cm/sec, what is the probability that she will get the ball in the hole? One high school student we had read these problems suggested: Would it get too complicated if they could vary both the initial velocity and the angle? That might be a possible, if very challenging, extension! KEYWORDS Two dimensional projectile motion, constant speed, parametric equations, coefficient of restitution, graphical analysis, numerical analysis. TEACHER NOTES ISSUES RELATED TO THE PROBLEM Prerequisites Projectile motion in two dimensions, parametric equations. Time allotment - time management Expectations Future payoffs Extensions A Variation of the Problem Another Approach: Fix the original velocity and vary the angle. The box is 40 cm on each side and 200 cm deep. The hole at the bottom of the box is on the far side and is 5 cm deep, running the length of the box, i.e. 40 cm. From the top of the box on the near side (opposite the far side! and at a point right in the middle of the nearside edge) contestants are permitted to aim/angle a spring loaded gun to fire a small (1 cm radius) rubber ball. The gun is fixed perpendicular to the nearside edge. The velocity which the gun imparts to the ball is 400 cm/sec and is fixed. The ball proceeds to bounce off the far side, then the near side, then the far side, etc. until it reaches the bottom of the box. The only variable contestants are permitted to control is the angle at which they shoot the ball. Question 1: Build a model of this phenomena, being sure to state your assumptions, and show that you can win every time by prescribing a suitable set of angles (between 0 and 90 degrees) at which you can fire the ball and land in the hole every time!! Question 2: If the typical user just selects a setting at random between 0 and 90 degrees, what is the probability that she will get the ball in the hole? References and Sources The idea came from Sample Problem 6, p. 61, in Fundamentals of Physics, Extended Third Edition, by David Halliday and Robert Resnick, John Wiley & Sons, Inc.: New York. 1988. POSSIBLE SOLUTION(S) Question 1: We define the variables: s = 40 cm is the width of the box; h = 200 cm is the height of the box; g = 981 cm/sec^2 is the acceleration due to gravity; and e = .7 is the coefficient of restitution. s = 40; h = 200; g = 981; e = .7; We construct a parametric function p[t,v0] of time (t) and original velocity (v0) through which we predict where the ball will be at time t in sec. We compute the time it takes to go from one side of the box to the other using distance/velocity = time, i.e. in the interval between firing the gun and hitting the first wall hit we have time T0 = 40/v0; in interval between hitting the wall for the first time and hitting the wall the second time we have time T1 = 40/(e v0) (where e is the coefficient of restitution), in interval between hitting the wall for the second time and hitting the wall the third time we have time T2 = 40/(e^2v0), etc. And we keep track of whether we are going left (we place our cannon on the left wall of the box) to right or right to left all the while noting the vertical position is h - 1/2 g t^2. This function p presumes that we shoot the ball horizontally and we have the option of varying the original velocity, v0, only. p[t_,v0_] := If[t<=s/v0,{v0 t,h - 1/2 g t^2}, (* Left to Right *) If[t <= s/v0(1 + 1/e), (* Right to Left *) {s - e v0 (t - s/v0), h - 1/2 g t^2}, If[t <= s/v0(1 + 1/e + 1/e^2), (* Left to Right *) {e^2 v0 (t-s/v0(1 + 1/e)), h - 1/2 g t^2}, If[t <= s/v0(1 + 1/e + 1/e^2 + 1/e^3), (* Right to Left *) {s - e^3 v0 (t-s/v0(1 + 1/e + 1/e^2)), h - 1/2 g t^2}, If[t <= s/v0(1 + 1/e +1/e^2 + 1/e^3 + 1/e^4), (* Left to Right *) {e^4 v0 (t - s/v0(1 + 1/e + 1/e^2 + 1/e^3)), h - 1/2 g t^2}, If[t <= s/v0(1 + 1/e + 1/e^2 + 1/e^3 + 1/e^4 + 1/e^5), (* Right to Left *) {s - e^5 v0 (t-s/v0(1 + 1/e + 1/e^2 + 1/e^3 + 1/e^4)), h - 1/2 g t^2}, If[t <= s/v0(1 + 1/e +1/e^2 + 1/e^3 + 1/e^4 + 1/e^5 + 1/e^6), (* Left to Right *) {e^6 v0 (t - s/v0(1 + 1/e + 1/e^2 + 1/e^3 + 1/e^4 + 1/e^5)), h - 1/2 g t^2}, If[t <= s/v0(1 + 1/e + 1/e^2 + 1/e^3 + 1/e^4 + 1/e^5 + 1/e^6 + 1/e^7), (* Right to Left *) {s - e^7 v0 (t-s/v0(1 + 1/e + 1/e^2 + 1/e^3 + 1/e^4 + 1/e^6)), h - 1/2 g t^2}, {s - e^7 v0 (t-s/v0(1 + 1/e + 1/e^2 + 1/e^3 + 1/e^4 + 1/e^6)), h - 1/2 g t^2}]]]]]]]]//N We offer up two plotting routines: r[v0,ti] permits us to examine the motion of the ball in the time interval [0, ti] with initial velocity v0 over the entire box, while q[v0,ti] . This function p[t,v0] will effectively plot 8 bounces only, thus even though one might prescribe a long time interval there will only be 8 bounces recorded. r[v0_,ti_] := ParametricPlot[{p[t,v0], {40,200 t/ti}},{t,0,ti}, PlotRange->{{0,40},{0,200}}] q[v0_,ti_] := ParametricPlot[{p[t,v0], {40,200 t/ti}},{t,0,ti}, PlotRange->{{30,40},{0,200}}] From inspection of graphs we see that the ball will land in the bottom hole if given initial horizontal velocities in the ranges (first time hit - but disallowed as the velocities are less than gun minimum of 100 cm/sec) 55-74 cm/sec, (second time hit) 265-304 cm/sec - which is our interval of choice, and (third time hit) 691-770 cm/sec which is beyond the range of the gun. These are not precise values, but close estimates from the graphs. We show the trajectories for the end points for these respective time intervals. Show[r[55,1],r[74,1]] Show[r[265,1],r[304,1]] Show[r[691,1],r[770,1]] Question 2: If then a game player merely selects a gun velocity at random from 100 to 500 cm.sec and only the velocity range 265 - 304 cm/sec will actually cause the ball to go through the hole then the probability that the the player wins is approximately 0.0975 or only 10%, i.e. one chance in ten. (304-265)/(500-100)//N 0.0975 Clearly closing the hole will decrease these chances and one might wish to consider how the width of the hole varies the chance of winning, i.e., does a linear increase in size of hole imply a linear increase in chance of winning? ISSUES IN SOLUTION This problem, beyond the simple application of the two dimensional projectile motion equations demands a good bookkeeping scheme in writing out equations for each interval between hits of the walls. We have offered a graphical analysis with a plotting tool to trial and error values which we found to be hard enough let alone consider an analytic solution.