Home Run Derby BRIEF ABSTRACT In this projectile (base ball) motion problem, we seek to find out whether a player makes a home run given the dimensions of the stadium field as well as the ball's initial velocity and launch angle. This could be used as a class room activity or homework problem. GENERAL INFORMATION FileName: HOMERUN Full title: Home Run Derby Last Update: 5/27/96 Developers: Robert Feller, Mathematics Department, Richmond High School, Richmond IN 47374 USA Kimberly Foltz, Mathematics and Computer Science Division, Indiana Academy for Science, Mathematics, & Humanities, Muncie IN 47306 USA Lynn Kiaer, Department of Mathematics, Rose-Hulman Institute of Technology, Terre Haute IN 47803 USA Aaron Klebanoff, Department of Mathematics, Rose-Hulman Institute of Technology, Terre Haute IN 47803 USA Contact: Aaron Klebanoff, Department of Mathematics, Rose-Hulman Institute of Technology, Terre Haute IN 47803 USA. Phone: 812-877-8151. Email: Aaron.Klebanoff@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 THE PROBLEM For a projectile with initial velocity v0 , initial height y0, and angle of elevation theta, the parametric equations x(t) = v0 Cos[ theta ] t y(t) = -16t^2 + v0 Sin[ theta ] t + y0 are used to describe the motion of the projectile. This model considers the force due to gravity, but it does not consider wind resistance. We will model the flight path of a baseball by the parametric equations x(t) = v0 Cos[ theta ] t - 12 t^2 y(t) = -16t^2 + v0 Sin[ theta ] t + 3 - 10 t , where the additional terms have been included to simulate the time in flight, the influence of the wind, and the height of the ball at impact (3 feet) . This model is still not quite physically correct, for resistance to wind is often regarded as proportional to velocity raised to some power yielding a different and more complex model. 1. Joe Homerun hits the ball with an initial velocity of 110 mph and at angle of elevation of 35 degrees. What is the total horizontal distance the ball travels before it hits the ground? Graph the path of the flight of the ball. 2. With bases loaded, a full count, and two out in the bottom of the ninth, Joe's team is down two runs. Joe makes contact with the next pitch giving it a 116 mph initial velocity and a 25 degree angle of elevation. The ball heads directly towards center field where the 10 foot fence is 404 ft from home plate. Did Joe hit a home run? 3. A baseball stadium is designed so that the playing field is the region bounded by the circle with equation (x-104)^2 + (y-104)^2 = 256^2 and the first quadrant. Home plate is located at the origin. If the ball is hit with an initial velocity of 110 mph and an angle of elevation of 30 degrees, describe the home run zone in terms of an acute angle, a, with vertex at home plate and which is measured from the first base line, i.e. a is 0 degrees at the first base line and 90 degrees at the third base line. Assume the fence is 10 feet tall. 4. Fenway Park in Boston is the home of the legendary Green Monster, a 37 foot high wall in left field. The distance from home plate to the outfield fence is 315 feet down the third baseline. Conventional baseball wisdom has it that this park favors right handed hitters because of this short distance to the wall as compared to other parks. For example, Riverfront Stadium in Cincinnati is 330 feet down the third baseline. Assume the fence is 10 feet tall. If a ball is hit down the third baseline with an initial velocity of 105 mph and an angle of elevation of 28 degrees, in which park(s) would it be a home run? By how much does the ball clear/not clear the fence in each park? KEYWORDS Projectile motion, parametric equations, trigonometric functions. TEACHER NOTES ISSUES RELATED TO THE PROBLEM Prerequisites This problem is doable once the students have been introduced to projectile motion and parametric equations assuming that they will be using a graphing utility and/or a computer algebra system. Time allotment - time management Expectations Future payoffs Extensions As a possible extension of this problem, you may want to have the students experiment with the values of the initial velocity and angle of inclination. Major differences occur with just a slight variation of one of these parameters. A discussion of other factors that affect the path of the ball could include the spin of the ball before and after impact, the point of impact of the ball with the bat, and the initial height of the ball. (This model assumes that the initial height of the ball at impact was 3 feet.) Further one could give the students the more realistic differential equation model with resistance and ask to estimate the parameters, e.g., in one dimension y'(t) = - g + a y(t) . References and Sources POSSIBLE SOLUTIONS PART 1 Enter the initial conditions and the equations for the model. In[2]:= vnaught = 110*5280/3600; (*converts mph to f/s*) theta = 35* Pi/180; (*converts degrees to radians*) In[4]:= horiz[t_]:=vnaught Cos[theta] t - 12t^2; vert[t_]:=-16t^2 + vnaught Sin[theta] t+ 3 - 10*t; In order to know the total horizontal distance traveled, we first must determine WHEN it will hit the ground. (The ball will hit the ground when its vertical position is 0.) We then will evaluate the horizontal position for this value of time . In[6]:= timeground = Solve[vert[t]==0,t] //N Out[6]= {{t -> -0.0360948}, {t -> 5.19466}} In[7]:= horiz[t] /. timeground[[2]] //N Out[7]= 362.694 Thus the total horizontal distance the ball travels before it hits the ground is 362.7 ft. The path of the ball will follow the graph of the parametric equations. It looks reasonable as the ball tends to "die" in the outfield. In[8]:= ParametricPlot[{horiz[t], vert[t]}, {t, 0, 10}, PlotRange -> {0, 150}, AspectRatio -> Automatic, PlotLabel -> "Height vs. Distance in feet"] Out[8]= -Graphics- PART 2 Change the initial conditions. In[9]:= vnaught = 116*5280/3600; theta = 25*Pi/180; One approach is to find when the ball has gone the desired distance and then see if it is still high enough at that point in time to clear the fence. In[11]:= timedist = Solve[horiz[t] == 404,t] //N Out[11]= {{t -> 3.66604}, {t -> 9.18339}} In[12]:= height=vert[t] /. timedist[[1]]//N Out[12]= 14.8957 A second approach is to find when the ball reaches the height of 10 feet. (This should happen exactly twice, once on the way up and once on the way down.) If the ball does not come back down until after it passes the 404 mark, then it cleared the fence. In[13]:= timeheight = Solve[vert[t] == 10,t] //N Out[13]= {{t -> 0.116597}, {t -> 3.75224}} In[14]:= dist = horiz[t] /. timeheight[[2]] //N Out[14]= 409.618 By either method, it's gone!!! Joe's team wins!!! PART 3 Change the initial conditions. In[15]:= vnaught = 110*5280/3600; theta = 30*Pi/180; The ball has a chance of going out only until its height comes back down to 10 feet. How far horizontally has it gone at this point? In[17]:= timeheight=Solve[vert[t]==10,t]//N Out[17]= {{t -> 0.101384}, {t -> 4.31528}} In[18]:= dist=horiz[t]/.timeheight[[2]]//N Out[18]= 379.466 After the ball has gone a horizontal distance of (approximately) 379.5 feet from home plate, it will no longer be high enough to clear the fence. Thus, it will be a home run only if it is directed toward those portions of the fence that are no more than 379.5 feet from home plate . So, e are interested in all points that are within 379.5 feet of home plate AND are out of the ballpark. This is illustrated by the intersection of the circle centered at home plate with a radius of 379.5 (shown in bold) and the circle for our home run range (shown by the dashed curve). In[19]:= p1 = ImplicitPlot[{x^2 + y^2 == dist^2, (x-104)^2 + (y-104)^2 == 256^2}, {x, 0, 380}, {y, 0, 380}, AspectRatio -> Automatic, PlotStyle -> {AbsoluteDashing[{3, 6}], AbsoluteThickness[2]}] Out[19]= -Graphics- By finding the points of intersection of these two circles and connecting these points to the origin, we will section the outfield fence into zones where the hit will be good. We can then use right triangles to determine the measurements of acute angles alpha which partition the outfield into home run zones. In[20]:= sol = NSolve[{x^2 + y^2 == dist^2, (x-104)^2 + (y-104)^2 == 256^2}, {x, y}] Out[20]= {{x -> 121.825, y -> 359.379}, {x -> 359.379, y -> 121.825}} In[21]:= ArcTan[y/x]*180/Pi /. sol[[2]] //N Out[21]= 18.7261 In[22]:= ArcTan[y/x]*180/Pi /. sol[[1]] //N Out[22]= 71.2739 Thus, measured from the first baseline, the approximate home run zone is between 0 and 18.7 degrees or between 71.3 degrees and 90 degrees. If this hit is directed into one of these zones, it will be a home run. We see this from the above combined diagram of field and limiting lines. Here's a nice way to visualize the homerun regions. WARNING: The contourplot takes a while to produce. To get the idea without a great time expense, decrease PlotPoints to about 50 in p4. In[23]:= m2 = y/x /. sol[[2]]; m1 = y/x /. sol[[1]]; In[25]:= p2 = Plot[{0, m1 x, m2 x}, {x, 0, 400}, PlotStyle -> AbsoluteThickness[3]] Out[25]= -Graphics- In[26]:= yints = Solve[(0-104)^2 + (y-104)^2 == 256^2, y] Out[26]= 208 - 48 Sqrt[95] 208 + 48 Sqrt[95] {{y -> -----------------}, {y -> -----------------}} 2 2 In[27]:= yint = y /. yints[[2, 1]] Out[27]= 208 + 48 Sqrt[95] ----------------- 2 In[28]:= p3 = ParametricPlot[{0, t}, {t, 0, yint}, PlotStyle -> AbsoluteThickness[3]] Out[28]= -Graphics- In[29]:= Clear[f, x, y] In[30]:= f[x_, y_] = If[(y < m2 x && x < Sqrt[dist^2 - y^2]) || (y > m1 x && y < Sqrt[dist^2 - x^2]), 0, 1]; In[31]:= p4 = ContourPlot[f[x, y], {x, 0, 400}, {y, 0, 400}, Contours -> 1, PlotPoints -> 200] Out[31]= -ContourGraphics- In[32]:= Show[p4, p1, p2, p3, PlotRange -> {{-1,400},{-1,400}}, PlotLabel -> "Home Run Zones in Black"] Out[32]= -Graphics- Thus, measured from the first baseline, the approximate home run zone is between 0 and 18.5 degrees or between 71.5 degrees and 90 degrees. If this hit is directed into one of these zones, it will be a home run. We see this from the above combined diagram of field and limiting lines. PART 4 Change the initial conditions and view the path. Any predictions? In[33]:= vnaught = 105*5280/3600; theta = 28* Pi/180; In[35]:= ParametricPlot[{horiz[t], vert[t]}, {t, 0, 5}, PlotRange -> {0,100}, AspectRatio -> Automatic, PlotLabel -> "Height vs. Distance in feet"] Out[35]= -Graphics- Find if the hit is good in Riverfront (as done in Part 2). In[36]:= timedist=Solve[horiz[t] == 330,t] //N Out[36]= {{t -> 3.5211}, {t -> 7.81006}} In[37]:= height = vert[t] /. timedist[[1]] //N Out[37]= 23.9894 When it gets to the wall, it still has the height to clear the fence. Now check on the same shot in Fenway Park. In[38]:= timetowall = Solve[horiz[t] == 315,t] //N Out[38]= {{t -> 3.24715}, {t -> 8.08401}} In[39]:= height = vert[t] /. timetowall[[1]] //N Out[39]= 36.5892 Not quite. Maybe Fenway isn't so friendly after all. The hit that was a home run in Riverfront just misses going out in Fenway. Do not generalize this too quickly. This may not always be the case. By adjusting the initial velocity or the angle of elevation, you might be able to reverse this scenario. ISSUES IN SOLUTION