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Differential Equations Physics for Game Programmers GRANT PALMER Physics for Game Programmers Copyright © 2005 by Grant Palmer All rights reserved. No part of this work may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage or retrieval system, without the prior written permission of the copyright owner and the publisher. ISBN (pbk): 1-59059-472-X Printed and bound in the United States of America 9 8 7 6 5 4 3 2 1 Trademarked names may appear in this book. Rather than use a trademark symbol with every occurrence of a trademarked name, we use the names only in an editorial fashion and to the benefit of the trademark owner, with no intention of infringement of the trademark. Lead Editor: Tony Davis Technical Reviewers: Alan McLeod, Jack Park Editorial Board: Steve Anglin, Dan Appleman, Ewan Buckingham, Gary Cornell, Tony Davis, Jason Gilmore, Jonathan Hassell, Chris Mills, Dominic Shakeshaft, Jim Sumser Assistant Publisher: Grace Wong Project Manager: Laura E. Brown Copy Manager: Nicole LeClerc Copy Editor: Ami Knox Production Manager: Kari Brooks-Copony Production Editor: Kelly Winquist Compositor: Susan Glinert Proofreader: Liz Welch Indexer: John Collin Artist: Kinetic Publishing Services, LLC Cover Designer: Kurt Krames Manufacturing Manager: Tom Debolski Distributed to the book trade in the United States by Springer-Verlag New York, Inc., 233 Spring Street, 6th Floor, New York, NY 10013, and outside the United States by Springer-Verlag GmbH & Co. KG, Tiergartenstr. 17, 69112 Heidelberg, Germany. In the United States: phone 1-800-SPRINGER, fax 201-348-4505, e-mail [email protected], or visit http://www.springer-ny.com. Outside the United States: fax +49 6221 345229, e-mail [email protected], or visit http://www.springer.de. For information on translations, please contact Apress directly at 2560 Ninth Street, Suite 219, Berkeley, CA 94710. Phone 510-549-5930, fax 510-549-5939, e-mail [email protected], or visit http://www.apress.com. The information in this book is distributed on an “as is” basis, without warranty. Although every precaution has been taken in the preparation of this work, neither the author(s) nor Apress shall have any liability to any person or entity with respect to any loss or damage caused or alleged to be caused directly or indirectly by the information contained in this work. The source code for this book is available to readers at www.apress.com in the Downloads section. This book is dedicated to my wonderful wife, Lisa. Contents at a Glance About the Author . xvii About the Technical Reviewers . xix Acknowledgments . xxi Introduction . xxiii CHAPTER 1 Adding Realism to Your Games . 1 CHAPTER 2 Some Basic Concepts . 5 CHAPTER 3 Basic Newtonian Mechanics . 21 CHAPTER 4 Basic Kinematics . 49 CHAPTER 5 Projectiles . 83 CHAPTER 6 Collisions . 139 CHAPTER 7 Sports Simulations . 167 CHAPTER 8 Cars and Motorcycles . 211 CHAPTER 9 Boats and Things That Float . 245 CHAPTER 10 Airplanes . 275 CHAPTER 11 Rockets and Missiles . 319 CHAPTER 12 The Physics of Solids . 351 CHAPTER 13 Explosions . 371 CHAPTER 14 Lasers . 385 CHAPTER 15 Probabilistic and Monte Carlo Simulations . 403 ■INDEX . 423 v Contents About the Author . xvii About the Technical Reviewers . xix Acknowledgments . xxi Introduction . xxiii ■CHAPTER 1 Adding Realism to Your Games . 1 Physics Will Keep Your Games from Looking Fake . 1 Adding Physics-Based Realism Is Easier Than You Might Think . 2 Adding Physics Won’t Affect Game Performance . 3 Knowing Some Physics Will Make You a Better Game Programmer . 4 Let’s Add Some Realism to Your Games . 4 ■CHAPTER 2 Some Basic Concepts . 5 Systems of Units . 6 Scientific Notation . 7 Summation Notation . 8 Greek Letters . 8 Coordinate Systems and Frames of Reference . 9 Scalars and Vectors . 10 Computing the Magnitude of a Vector . 11 The Unit Vector . 11 Vector Cross Product . 12 Matrices . 12 Matrix Multiplication. 13 Rotation Matrices . 13 Derivatives . 15 Differential Equations . 18 Summary . 19 vii viii ■CONTENTS ■CHAPTER 3 Basic Newtonian Mechanics . 21 A Short Biography of Sir Isaac Newton . 22 Newton’s First Law of Motion: Inertia. 22 Newton’s Second Law of Motion: Force, Mass, and Acceleration . 23 Newton’s Third Law of Motion: Equal and Opposite Forces . 24 Force Vector . 24 Types of Forces . 25 Gravitational Force . 25 Friction . 31 Centripetal Force. 38 Force Balances and Force Diagrams . 39 Work . 41 Energy . 43 Kinetic Energy . 43 Potential Energy . 44 Other Forms of Energy . 44 Conservation of Energy . 45 Power . 45 Summary . 46 Answers to Exercises . 46 ■CHAPTER 4 Basic Kinematics . 49 Translational Motion . 49 The Relationship Between Force, Acceleration, Velocity, and Location . 50 Solving the Translational Equations of.
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