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### tennis_ball

Course: MATH 715473, Fall 2009
School: CSU Northridge
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Word Count: 430

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382: Math Scientic Computing Trajectory of a Spinning Tennis Ball Goal: This hand out formulates an ODE model to describe the trajectory of a spinning tennis ball and discusses how to calculate its solution under different assumptions (e.g., vacuum vs. air resistance) with MATLAB. The Model m mass of the ball angular velocity, a vector in the direction of the axis of rotation. 1 consult gure presented in...

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382: Math Scientic Computing Trajectory of a Spinning Tennis Ball Goal: This hand out formulates an ODE model to describe the trajectory of a spinning tennis ball and discusses how to calculate its solution under different assumptions (e.g., vacuum vs. air resistance) with MATLAB. The Model m mass of the ball angular velocity, a vector in the direction of the axis of rotation. 1 consult gure presented in class regard ball as a mass point (why is this ok? after all, we are considering the effects of the air in the motion of the ball, size and surface should matter... we are not really ignoring those...) moving under the inuence of the forces: weight drag G = mg, g = (0, 0, g) D = DL (v) v , v opposite to velocity orthogonal to and v Magnus force M = ML v , v The magnitudes of the drag and Magnus forces are given by 1 d2 2 v 2 4 1 d2 2 v 2 4 DL (v) = CD ML (v) = CM density of air (1) (2) CD , CM - determined experimentally, they depend on velocity, ball revolution and material of surface (it is ok to assume point mass) Experimental data for tennis ball suggests: Math 382: Scientic Computing 2 CD = 0.508 + 1 22.053 + 4.196 v w 5 2 2 5 (3) CM = where w = d 2 1 2.022 + 0.981 v w (4) | v | v Newtons second law trajectory denes of the ball v v d2 r(t) = mg DL (v) + ML (v) m 2 dt v v with initial conditions r(0) = r0 and dr (0) = v0 dt (5) Most usual case is the topspin lob lies on horizontal plane and is orthogonal to v0 (so by equation (5) also orthogonal to v(t)). This means motion is on vertical plane, so we chose x-axis to run along that plane, (5) becomes x = CD v x + CM v z z = g CD v z CM v x d2 , 8m (6) (7) where v = x2 + z 2 and = is direction of rotation, = 1 for topspin Initial conditions: x(0) = 0, z(0) = h, x(0) = v0 cos , z(0) = v0 sin (8) where v0 is the magnitude of the initial velocity vector v0 and the angle between v0 and the x-axis. Case I: motion in vacuum x = 0, with the above initial condi...

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CSU Northridge - MATH - 150
MATH 150ACalculus IFall 2007Review Sheet for Midterm Exam III Problem 1. Determine whether the following statements are True or False. Justify your answer.3(a)3(ax2 + bx + c) dx = 203(ax2 + c) dx.(b) If f is continuous on [a, b], t
CSU Northridge - MATH - 715473
MATH 150ACalculus IFall 2007Review Sheet for Midterm Exam III Problem 1. Determine whether the following statements are True or False. Justify your answer.3(a)3(ax2 + bx + c) dx = 203(ax2 + c) dx.(b) If f is continuous on [a, b], t
CSU Northridge - MATH - 150
MATH 150ACalculus IFall 2007Review Sheet for Midterm Exam II HW #8, due Thurs. Nov. 1st, 2007: Turn in solutions for problems 1 (a), (e), (m), and (n); 4; 6; 9; 11; and 13. Problem 1. Determine whether the following statements are True or False
CSU Northridge - MATH - 715473
MATH 150ACalculus IFall 2007Review Sheet for Midterm Exam II HW #8, due Thurs. Nov. 1st, 2007: Turn in solutions for problems 1 (a), (e), (m), and (n); 4; 6; 9; 11; and 13. Problem 1. Determine whether the following statements are True or False
CSU Northridge - MATH - 396
MATH 396Math for 3D GraphicsFall 2008Review Sheet for Midterm Exam The midterm exam will take place on Oct. 23rd during regular class time at our usual location, JR254. You are allowed to bring one page (one side of an 8.5 11 in. sheet of pape
CSU Northridge - MATH - 715473
MATH 396Math for 3D GraphicsFall 2008Review Sheet for Midterm Exam The midterm exam will take place on Oct. 23rd during regular class time at our usual location, JR254. You are allowed to bring one page (one side of an 8.5 11 in. sheet of pape
CSU Northridge - MATH - 255
MATH 255Applied Honors Calculus III Review Sheet for 2nd Midterm Exam Part IIWinter 20051. True or False. Determine whether the following statements are true or false and provide an explanation for your answer (5 points each). (a) If a particl
CSU Northridge - MATH - 715473
MATH 255Applied Honors Calculus III Review Sheet for 2nd Midterm Exam Part IIWinter 20051. True or False. Determine whether the following statements are true or false and provide an explanation for your answer (5 points each). (a) If a particl
CSU Northridge - MATH - 382
MATH 382Scientic ComputingFall 2008Final Project Spinning Tennis Ball Due: Tue. Dec. 16, 2008 by 5.00 p.m. Project Description: For the course nal project you need to solve the system of ODEs x = CD v x + CM v z z = g CD v z CM v x.
CSU Northridge - MATH - 715473
MATH 382Scientic ComputingFall 2008Final Project Spinning Tennis Ball Due: Tue. Dec. 16, 2008 by 5.00 p.m. Project Description: For the course nal project you need to solve the system of ODEs x = CD v x + CM v z z = g CD v z CM v x.
CSU Northridge - MATH - 396
MATH 396Math for 3D GraphicsFall 2008Lab Assignment # 2 matlab Functions Due: Thurs. Sep. 18, 2008 Directions: In this assignment youll practice writing matlab functions and scripts. You may want to read the lab notes for Thursday Sept. 11. In
CSU Northridge - MATH - 715473
MATH 396Math for 3D GraphicsFall 2008Lab Assignment # 2 matlab Functions Due: Thurs. Sep. 18, 2008 Directions: In this assignment youll practice writing matlab functions and scripts. You may want to read the lab notes for Thursday Sept. 11. In
CSU Northridge - MATH - 255
MATH 255Applied Honors Calculus III Final Exam Review Sheet Part IWinter 20051. True or False. Determine whether the following statements are true or false and provide an explanation for your answer (5 points each). (a)1 0 1ex12 +y 2si
CSU Northridge - MATH - 715473
MATH 255Applied Honors Calculus III Final Exam Review Sheet Part IWinter 20051. True or False. Determine whether the following statements are true or false and provide an explanation for your answer (5 points each). (a)1 0 1ex12 +y 2si
CSU Northridge - MATH - 382
MATH 382Scientic ComputingSpring 2009Lab Assignment # 3 Koch Curve Due: Tue. Mar. 17, 2009 Directions: In this assignment youll practice implementing more recursive algorithms in matlab. You can work with others and discuss the problems, but e
CSU Northridge - MATH - 715473
MATH 382Scientic ComputingSpring 2009Lab Assignment # 3 Koch Curve Due: Tue. Mar. 17, 2009 Directions: In this assignment youll practice implementing more recursive algorithms in matlab. You can work with others and discuss the problems, but e
CSU Northridge - MATH - 396
math 396Lmath for 3D graphicsFall 08 - week onelab rulesgeneral rules: no food no drinks(i.e., what you are not supposed to do at the lab)don't print more than 10 pages respect others working at the lab - keep noise to a minimum care for lab
CSU Northridge - MATH - 715473
math 396Lmath for 3D graphicsFall 08 - week onelab rulesgeneral rules: no food no drinks(i.e., what you are not supposed to do at the lab)don't print more than 10 pages respect others working at the lab - keep noise to a minimum care for lab
CSU Northridge - MATH - 450
MATH 450Advanced Mathematics for Engineers I Review Sheet #1 for Final ExamFall 2006Problem 1. Determine whether the following statements are True or False. Justify your answer. (a) The transformation w = (c) The Laplace equation w = 3iz2 . z+5
CSU Northridge - MATH - 715473
MATH 450Advanced Mathematics for Engineers I Review Sheet #1 for Final ExamFall 2006Problem 1. Determine whether the following statements are True or False. Justify your answer. (a) The transformation w = (c) The Laplace equation w = 3iz2 . z+5
CSU Northridge - MATH - 250
MATH 250Calculus III Review Sheet for 2nd Midterm ExamSpring 2008HW#7: Turn in problems 1 (a), (c), (h), and (j); 5; 6; and 9. Due: Thurs. April 3 1. True or False. Determine whether the following statements are true or false and provide an exp
CSU Northridge - MATH - 715473
MATH 250Calculus III Review Sheet for 2nd Midterm ExamSpring 2008HW#7: Turn in problems 1 (a), (c), (h), and (j); 5; 6; and 9. Due: Thurs. April 3 1. True or False. Determine whether the following statements are true or false and provide an exp
CSU Northridge - MATH - 450
MATH 450Advanced Mathematics for Engineers I Review Sheet #1 for Midterm ExamFall 2006Problem 1. Determine whether the following statements are True or False. Justify your answer. (a) If y1 (x) and y2 (x) are two solutions of y + 2y + y = 0, x
CSU Northridge - MATH - 715473
MATH 450Advanced Mathematics for Engineers I Review Sheet #1 for Midterm ExamFall 2006Problem 1. Determine whether the following statements are True or False. Justify your answer. (a) If y1 (x) and y2 (x) are two solutions of y + 2y + y = 0, x
CSU Northridge - MATH - 581
MATH 581Numerical Methods for Linear SystemsSpring 2008Homework 3 Due: Thurs. March 6, 2008 Lecture 10, pg. 76: 10.1, 10.2, 10.3 Lecture 11, pg. 85: 11.1, 11.3 Additional Problem 1. For a given matrix A Rmn , with m n: (a) Find the orthogonal
CSU Northridge - MATH - 715473
MATH 581Numerical Methods for Linear SystemsSpring 2008Homework 3 Due: Thurs. March 6, 2008 Lecture 10, pg. 76: 10.1, 10.2, 10.3 Lecture 11, pg. 85: 11.1, 11.3 Additional Problem 1. For a given matrix A Rmn , with m n: (a) Find the orthogonal
CSU Northridge - MATH - 581
MATH 581Numerical Methods for Linear SystemsSpring 2008Homework 4 Due: Tue. April 15, 2008 1. Write matlab code that implements algorithms 20.1 and 21.1. Find the LU decomposition of the matrices A= 0 1 1 1 B= 1020 1 1 1and the matrix in exam
CSU Northridge - MATH - 715473
MATH 581Numerical Methods for Linear SystemsSpring 2008Homework 4 Due: Tue. April 15, 2008 1. Write matlab code that implements algorithms 20.1 and 21.1. Find the LU decomposition of the matrices A= 0 1 1 1 B= 1020 1 1 1and the matrix in exam
CSU Northridge - MATH - 250
MATH 250Calculus IIISpring 2008Homework 8 Due: Thurs. April. 17, 2008 Section 13.1, pg. 679: 1, 5, 9, 17, 21, 23, 29. Section 13.2, pg. 683: 1, 5, 11, 15, 17, 19, 27, 33, 34, 35. Section 13.3, pg. 689: 3, 9, 13, 15, 31, 33, 37, 41.
CSU Northridge - MATH - 715473
MATH 250Calculus IIISpring 2008Homework 8 Due: Thurs. April. 17, 2008 Section 13.1, pg. 679: 1, 5, 9, 17, 21, 23, 29. Section 13.2, pg. 683: 1, 5, 11, 15, 17, 19, 27, 33, 34, 35. Section 13.3, pg. 689: 3, 9, 13, 15, 31, 33, 37, 41.
CSU Northridge - MATH - 250
MATH 250Calculus IIISpring 2008Homework 9 Due: Thurs. April. 24, 2008 Section 13.4, pg. 695: 1, 3, 13, 19, 21, 33. Section 13.5, pg. 699: 3, 5, 7, 19. Section 13.7, pg. 711: 1, 3, 9, 11, 17.
CSU Northridge - MATH - 715473
MATH 250Calculus IIISpring 2008Homework 9 Due: Thurs. April. 24, 2008 Section 13.4, pg. 695: 1, 3, 13, 19, 21, 33. Section 13.5, pg. 699: 3, 5, 7, 19. Section 13.7, pg. 711: 1, 3, 9, 11, 17.
CSU Northridge - MATH - 250
MATH 250Calculus IIISpring 2008Homework 4 Due: Thurs. Feb. 28, 2008 Bzier Curves e Bzier Curves are often employed in computer aided design. These parametric curves are e dened by x(t) = x0 (1 t)3 + 3x1 t(1 t)2 + 3x2 t2 (1 t) + x3 t3 y(t) =
CSU Northridge - MATH - 715473
MATH 250Calculus IIISpring 2008Homework 4 Due: Thurs. Feb. 28, 2008 Bzier Curves e Bzier Curves are often employed in computer aided design. These parametric curves are e dened by x(t) = x0 (1 t)3 + 3x1 t(1 t)2 + 3x2 t2 (1 t) + x3 t3 y(t) =
CSU Northridge - MATH - 250
MATH 250Calculus IIISpring 2008Homework 11 Due: Thurs. May 15, 2008 Section 14.1, pg. 735: 1, 3, 7, 9, 13, 17, 21. Section 14.2, pg. 740: 1, 3, 5, 7, 17, 19, 21. Section 14.3, pg. 747: 1, 3, 5, 7, 13, 15, 21, 26.
CSU Northridge - MATH - 715473
MATH 250Calculus IIISpring 2008Homework 11 Due: Thurs. May 15, 2008 Section 14.1, pg. 735: 1, 3, 7, 9, 13, 17, 21. Section 14.2, pg. 740: 1, 3, 5, 7, 17, 19, 21. Section 14.3, pg. 747: 1, 3, 5, 7, 13, 15, 21, 26.
CSU Northridge - MATH - 250
MATH 250Calculus IIISpring 2008Homework 3 Due: Thurs. Feb. 14, 2008 Directions: For problems with multiple parts, do parts (a) and (c) only Section 11.6, pg. 592: Every odd. Section 11.7, pg. 601: 1, 5, 9, 13, 27, 33, 35, 43, 57, 61, 77.
CSU Northridge - MATH - 715473
MATH 250Calculus IIISpring 2008Homework 3 Due: Thurs. Feb. 14, 2008 Directions: For problems with multiple parts, do parts (a) and (c) only Section 11.6, pg. 592: Every odd. Section 11.7, pg. 601: 1, 5, 9, 13, 27, 33, 35, 43, 57, 61, 77.
CSU Northridge - MATH - 150
MATH 150ACalculus IFall 2007Homework 5 Due: Tue. Oct. 9, 2007 Section 3.6, pg. 169: 1, 3, 11, 15, 17, 19, 21, 29, 35, 51. Section 3.7, READ the examples in the section carefully, make sure you understand them (ask me if anything is unclear), th
CSU Northridge - MATH - 715473
MATH 150ACalculus IFall 2007Homework 5 Due: Tue. Oct. 9, 2007 Section 3.6, pg. 169: 1, 3, 11, 15, 17, 19, 21, 29, 35, 51. Section 3.7, READ the examples in the section carefully, make sure you understand them (ask me if anything is unclear), th
CSU Northridge - MATH - 255
MATH 255Applied Honors Calculus IIIWinter 2005Homework 11 Due: Wed. Apr. 20, 2005 Section 17.7, pg. 1155: 5, 9, 11, 13, 19, 24, 33. Section 17.8, pg. 1161: 3, 7, 13, 15, 17, 19. Section 17.9, pg. 1168: 3, 7, 19, 25, 27.
CSU Northridge - MATH - 715473
MATH 255Applied Honors Calculus IIIWinter 2005Homework 11 Due: Wed. Apr. 20, 2005 Section 17.7, pg. 1155: 5, 9, 11, 13, 19, 24, 33. Section 17.8, pg. 1161: 3, 7, 13, 15, 17, 19. Section 17.9, pg. 1168: 3, 7, 19, 25, 27.
CSU Northridge - MATH - 255
MATH 255Applied Honors Calculus IIIWinter 2005Homework 10 Due: Wed. Apr. 13, 2005 Section Section Section Section 17.3, 17.4, 17.5, 17.6, pg. pg. pg. pg. 1117: 1125: 1132: 1142: 3, 3, 3, 3, 9, 11, 19, 29, 33. 9, 13, 21, 25, 29. 12, 13, 31. 17,
CSU Northridge - MATH - 715473
MATH 255Applied Honors Calculus IIIWinter 2005Homework 10 Due: Wed. Apr. 13, 2005 Section Section Section Section 17.3, 17.4, 17.5, 17.6, pg. pg. pg. pg. 1117: 1125: 1132: 1142: 3, 3, 3, 3, 9, 11, 19, 29, 33. 9, 13, 21, 25, 29. 12, 13, 31. 17,
CSU Northridge - MATH - 150
MATH 150ACalculus IFall 2007Homework 1 Due: Tue. Sept. 4, 2007 Section 2.2, pg. 74: 5, 7, 9, 13, 15, 29, 40. Section 2.3, pg. 84: 2 (b), (e), and (f), 5, 11, 23, 25, 29, 37, 43.
CSU Northridge - MATH - 715473
MATH 150ACalculus IFall 2007Homework 1 Due: Tue. Sept. 4, 2007 Section 2.2, pg. 74: 5, 7, 9, 13, 15, 29, 40. Section 2.3, pg. 84: 2 (b), (e), and (f), 5, 11, 23, 25, 29, 37, 43.
CSU Northridge - MATH - 150
MATH 150ACalculus IFall 2007Homework 7 Due: Tue. Oct. 23, 2007 For each of the functions below: I. nd its domain and range II. nd its x- and y-axis intercepts III. determine whether the graph is symmetric with respect to the y-axis or the origi
CSU Northridge - MATH - 715473
MATH 150ACalculus IFall 2007Homework 7 Due: Tue. Oct. 23, 2007 For each of the functions below: I. nd its domain and range II. nd its x- and y-axis intercepts III. determine whether the graph is symmetric with respect to the y-axis or the origi
CSU Northridge - MATH - 150
MATH 150ACalculus IFall 2007Homework 4 Due: Tue. Sept. 25, 2007 Section 3.3, pg. 145: 19, 25, 31, 37, 49, 59, 63, 95. Section 3.4, pg. 154: 1 15 (odd), 23, 33, 35, 37, 39, 45, 47. Section 3.5, pg. 161: 1, 5, 13, 21, 41, 53.
CSU Northridge - MATH - 715473
MATH 150ACalculus IFall 2007Homework 4 Due: Tue. Sept. 25, 2007 Section 3.3, pg. 145: 19, 25, 31, 37, 49, 59, 63, 95. Section 3.4, pg. 154: 1 15 (odd), 23, 33, 35, 37, 39, 45, 47. Section 3.5, pg. 161: 1, 5, 13, 21, 41, 53.
CSU Northridge - MATH - 150
MATH 150ACalculus IFall 2007Homework 6 Due: Tue. Oct. 16, 2007 Section 3.9, pg. 193: 1, 3, 11, 17, 21, 31, 33, 35, 37. Section 4.1, pg. 211: 3, 5, 11, 17, 21, 25, 31, 35, 47, 53, 69. Section 4.2, pg. 219: 1, 5, 11, 19, 29, 33.
CSU Northridge - MATH - 715473
MATH 150ACalculus IFall 2007Homework 6 Due: Tue. Oct. 16, 2007 Section 3.9, pg. 193: 1, 3, 11, 17, 21, 31, 33, 35, 37. Section 4.1, pg. 211: 3, 5, 11, 17, 21, 25, 31, 35, 47, 53, 69. Section 4.2, pg. 219: 1, 5, 11, 19, 29, 33.
CSU Northridge - MATH - 150
MATH 150ACalculus IFall 2007Homework 3 Due: Tue. Sept. 18, 2007 Section 3.1, pg. 119: 1, 3, 5, 7, 11, 17, 27, 43. Section 3.2, pg. 131: 3, 5, 7, 17, 23, 39, 51. Section 3.3, pg. 145: 1 11 every odd.
CSU Northridge - MATH - 715473
MATH 150ACalculus IFall 2007Homework 3 Due: Tue. Sept. 18, 2007 Section 3.1, pg. 119: 1, 3, 5, 7, 11, 17, 27, 43. Section 3.2, pg. 131: 3, 5, 7, 17, 23, 39, 51. Section 3.3, pg. 145: 1 11 every odd.
CSU Northridge - MATH - 250
MATH 250Calculus IIISpring 2008Homework 6 Due: Thurs. Mar. 13, 2008 Section 12.4, pg. 640: 1, 3, 7, 11, 15, 17, 19, 21. Section 12.5, pg. 646: 5, 7, 9, 11, 13, 19, 21, 25. Section 12.6, pg. 651: 1, 3, 5, 9, 13, 19, 21, 31.
CSU Northridge - MATH - 715473
MATH 250Calculus IIISpring 2008Homework 6 Due: Thurs. Mar. 13, 2008 Section 12.4, pg. 640: 1, 3, 7, 11, 15, 17, 19, 21. Section 12.5, pg. 646: 5, 7, 9, 11, 13, 19, 21, 25. Section 12.6, pg. 651: 1, 3, 5, 9, 13, 19, 21, 31.
CSU Northridge - MATH - 150
MATH 150ACalculus IFall 2007Homework 9 Due: Thurs. Nov. 15, 2007 Section 5.1, pg. 299: 3, 17, 20, 21. Section 5.2, pg. 310: 3, 11, 17, 21, 29, 35, 53. Section 5.3, pg. 321: 5, 7, 19 25 (odds), 35. Additional Problems: 1. Prove:n n n(ai + bi
CSU Northridge - MATH - 715473
MATH 150ACalculus IFall 2007Homework 9 Due: Thurs. Nov. 15, 2007 Section 5.1, pg. 299: 3, 17, 20, 21. Section 5.2, pg. 310: 3, 11, 17, 21, 29, 35, 53. Section 5.3, pg. 321: 5, 7, 19 25 (odds), 35. Additional Problems: 1. Prove:n n n(ai + bi
CSU Northridge - MATH - 250
MATH 250Calculus IIISpring 2008Homework 5 Due: Thurs. Mar. 6, 2008 Section 12.1, pg. 622: 5, 21, 23, 25, 27, 33. Section 12.2, pg. 628: 1, 3, 5, 9, 27, 29, 33, 41. Section 12.3, pg. 634: 3, 15, 17, 19, 25, 27, 35.
CSU Northridge - MATH - 715473
MATH 250Calculus IIISpring 2008Homework 5 Due: Thurs. Mar. 6, 2008 Section 12.1, pg. 622: 5, 21, 23, 25, 27, 33. Section 12.2, pg. 628: 1, 3, 5, 9, 27, 29, 33, 41. Section 12.3, pg. 634: 3, 15, 17, 19, 25, 27, 35.
CSU Northridge - MATH - 150
MATH 150ACalculus IFall 2007Homework 2 Due: Tue. Sept. 11, 2007 Section 2.3, pg. 84: 47, 55. Section 2.4, pg. 95: 1, 11, 13, 15, 17 read section before attempting problems! Section 2.5, pg. 105: 1, 5, 7, 9, 13, 37, 41, 61, 62.