423

INDEX

■Aacceleration

see also angular acceleration; gravitational acceleration; rotational acceleration; translational acceleration

car racing simulations, 217, 222derivatives, 17, 18English System unit of, 23equations of motion, 53force and, 50gravitational force, 26Kawasaki 800 SX-R, 269location and, 51modeling for boats, 258–261Newton’s second law of motion, 23SI unit of, 23velocity and, 50

acceleration equationsaerodynamic drag, 105–106car racing simulations, 218gravity-only model, 85

actionPerformed methodBeanBag class, 54CarSimulator class, 234FreeKick class, 198GameUpdater class, 95GasTankSimulator class, 367GolfGame class, 181GolfGame2 class, 112GolfGame3 class, 120GolfGame4 class, 131GravityGame class, 29, 30LaserSimulator class, 398PaddleGame class, 155PiEstimator class, 419RocketSimulator class, 340Shuffleboard class, 35SoldierGame class, 416SphereCollision class, 148SpringSimulator class, 68

aerodynamic centerairfoils, airplanes, 310

aerodynamic dragairplanes, 278altitude effects, rockets, 331boats, 258car racing simulations, 220golf balls, 168gravitational force, 31projectile motion, 98–114

acceleration equations, 105–106altitude effects, 103drag coefficient, 98, 99, 103DragProjectile class, 108–111equations of motion, 104–105force diagram, 105force equations, 105–106friction drag, 98Golf Game, 111–113laminar flow, 103–104location equations, 107pressure drag, 98summary, 113–114terminal velocity, 108turbulent flow, 103–104velocity equations, 106–107wind effects on, 114–122

aerodynamics, airplanes, 295–301airplane orientation, 297center of gravity, 297climb angle, 296ground effect, 300landing, 301pitch angle, 296takeoff, 299turning, 296–297

aeroplanessee airplanes

aileron, airplanes, 277

424 ■I N D E X

airdensity of air and water, 249

Airborne Laser (ABL), 393–394airfoils, airplanes

aerodynamic center, 310angle of attack, 279camber line, 278chord line, 278creating lift, 279evaluating airfoil lift, 280Flight Simulator, 306incidence angle, 279leading edge, 278lift, 278

center of pressure, 284trailing edge, 278

airplane terminology, 276–277aileron, 277airfoil, 278aspect ratio, 277banking, 297center of pressure, 284chord, 277elevator, 277flaps, 277, 283fuselage, 276horizontal stabilizer, 277landing gear, 277lift, 278moments, 310pitch, 277rudder, 277span, 277stability, 313stall, 282tail, 276trim, 311vertical fin, 277vertical stabilizer, 277wing, 276yaw, 277

airplanes, 275–317aerodynamic drag, 278aerodynamics, 295–301airplane efficiency factor, 293airplane orientation, 297drag, 292–295Flight Simulator, 301–310force diagram, 278historical background to flying, 276

lift, 278–285references for further reading, 317thrust, 278, 285–292types of airplane engines, 285

air-to-air missiles, 347air-to-surface missiles, 347altitude effects

aerodynamic drag, 103gravitational acceleration, 330propeller engine thrust, 288

power drop-off factor, 290Rocket Simulator, 335rockets

aerodynamic drag, 331pressure and pressure ratio, 326thrust, 326

ammonium nitratereferences for further reading, 383TNT equivalence factor, 379types of explosives, 374

angle of attackairfoils, airplanes, 279

angular accelerationrolling motion, 76, 77rotational motion, 70torque, 72

bowling ball kinematics, 78angular impulse of torque, 159angular momentum, 159–160

conservation of, 160equation for, 159

angular velocitybowling ball kinematics, 78car racing simulations, 216, 218

driving around curves, 238radians, 70rolling motion, 76, 77rotational motion, 70

apparent slip ratiopropellers, 253

apparent velocitywind effects on projectile motion, 114

armor penetrationballistic impacts on animals, 356ballistic impacts on body armor, 354ballistic impacts on steel armor, 353bullets, 354references for further reading, 370Thompson “F-Formula”, 353

425■I N D E X

arrayssee matrices

arrowsballistic impacts, 351

on animals, 356body wounds, 358Monte Carlo simulations, 411projectile motion, 136–137

aspect ratio, airplanes, 277atmospheric measurements, calculating, 327

altitude effects, 332USatm76 class, 338

atoms, 386–389autoignition temperature, 372automatic transmission

car racing simulations, 220automobile

brief history of, 212axis of rotation

coordinate axes, 75for common objects, 73rolling motion, 76rotational motion, 71

■BBacon, Roger, 319ballistic impacts, 351–358

animals, 356body armor, 356effect of, 352energy considerations, 352–353momentum, 356–357steel armor, 354target penetration, 352

body armor, 354steel armor, 353

wounded bodies, 357–358banking, airplanes, 297baseball

brief history of, 204baseball simulation, 204–207

equipment specifications, 204–205modeling bat hitting ball, 207modeling pitching, 206–207

drag coefficient of baseball, 207effects of spin on flight of baseball, 206

references for further reading, 210basketball

brief history of, 199

basketball simulation, 199–204equipment specifications, 199–201free-throw game, 203–204modeling jump shot, 202–203

determining if shot is good, 203forces on basketball in flight, 202modeling backboard impacts, 203

beam, boats, 246Beanbag Game

translational equations of motion, 53Bernoulli’s equation, 123

airfoils creating lift, 280blast damage

blast wave, 375calculating damage

other explosives (not TNT), 378–379TNT equivalence factors, 379TNT explosion model, 376–378

explosions, 375–380overpressure, 375personal injury, 376structural damage, 375TNT equivalence model, 378

blast wave, 372blast damage, 375

blunt traumamomentum in ballistic impacts, 357

Boat Simulatorsee speedboat simulations

boatsboat hull types, 246–247boat thrust, 253

apparent slip ratio, 253drive systems, 250propellers, 251wake velocity, 252

force diagram, 247–248forces acting on boats, 248

aerodynamic drag, 258buoyancy, 248–250hydrodynamic drag, 254–257thrust, 250–254

geometry terminology, 246jet ski simulations, 268–269modeling acceleration and velocity,

258–261using performance data, 258

426 ■I N D E X

nautical terminology, 245beam, 246bow, 246displacement, 246draft, 246hull, 246jib, 269keel, 269knot, 246mainsail, 269mast, 269port, 246starboard, 246stern, 246transom, 246

physics of sailing, 269–270physics of surfing, 270–273references for further reading, 273speedboat simulations, 261–267velocity and propeller pitch, 252

BoatSimulator class, 265resetDisplay method, 266

body armorballistic impacts on, 354NIJ classification standards, 355

body woundsballistic impacts, 357

bombsmodeling fragmentation devices, 380TNT equivalence of bombs, 379–380

boundary conditions, 59bow, boats, 246bowling ball kinematics

rigid body motion, 77BoxsterS class

car racing simulations, 233bullets

armor penetration, 354ballistic impacts, 351–358body wounds, 357characteristics of bullets, 135drag coefficient, 135momentum in ballistic impacts, 357NIJ body armor tests, 355projectile motion, 134–135

Thompson “F-Formula”, 354yaw angle, 134

buoyancy, boats, 248–250buoyancy force, 249density and, 250density of air and water, 249forces acting on boats, 248surfing, 271

■CC4 (Composition C-4)

TNT equivalence factor, 379types of explosives, 374

camber lineairfoils, airplanes, 278

cannonballsprojectile motion, 135–136

Car classcar racing simulations, 228getRightHandSide method, 231

car racing simulations, 212–240aerodynamic drag, 220automatic transmission, 220Car class, 228car simulation, 227drag coefficient, 221drag force, 221driving around curves, 237

high-speed turns, 238engine braking, 226engine torque, 214, 216force diagram, 212gear ratio, 217gear shifting, 219gears and wheel Torque, 217manual transmission, 220modeling car crashes, 239motorcycle racing compared, 240power, 216references, 243rolling friction, 221wheel radius, 219wheel traction, 236

carsbrief history of, 212

CarSimulator class, 234Cartesian coordinate system, 9

computing magnitude of vector, 11conversion to spherical system, 10usage of, 25

427■I N D E X

center of gravityaerodynamics, airplanes, 297

center of massrigid body motion, 73–74rolling motion, 76sailboat, 270surfing, 271

center of rotationmoments, 310

center of pressureairfoils, airplane lift, 284

centripetal force, 38–39physics and programming, 4Rocket Simulator, 336satellites, 342summarized, 46

Cessna 172 Skyhawkairplane drag, 295airplane thrust, 289drag profiles, 294Flight Simulator, 302

CessnaSkyhawk classFlight Simulator, 309

chaff, missile guidance, 348chemical energy, 43, 44chemical explosions, 371chord, airplanes, 277

chord line, 278circular motion, 10circular orbits, 343climb angle, 296closed-form solution

integration, 52coefficient of drag

see drag coefficientcoefficient of friction, 32, 34

counterbalancing forces, 33friction coefficients of common surfaces, 34frictional force, 46kinetic coefficient of friction, 34pushing/pulling a box, 42rolling motion, 76Shuffleboard Game, 34static coefficient of friction, 34

coefficient of propulsive efficiencyboat thrust, 253

coefficient of resistancedrag force, boats, 254

coefficient of restitutionbasketball ball-backboard impact, 203golf balls, 168two-body linear collisions, 145

coefficient of rolling frictioncar racing simulations, 222golf simulation, 188, 189

coherent light, 389collisions, 139–165

angular momentum, 159–160baseball bat hitting ball, 207basketball ball-backboard impact, 203car racing simulations, 239determining occurrence of, 157–159direct impact collisions, 160elastic and inelastic collisions, 142–143friction and, 160–164golf club-ball impact, 171, 174–175

putting, 188immovable objects, 146kinetic energy, 142line of action, 144linear momentum, 140–141

conservation of, 141–142modeling car crashes, 239oblique collisions, 160

three-dimensional, 164two-dimensional, 162

soccer ball-foot impact, 191three-dimensional collisions, 157, 164two-body linear collisions, 143–149

linear collision simulator, 146post-collision velocity equations, 144

two-dimensional collisions, 149–157, 162computational fluid dynamics

evaluating airfoil lift, 280conservation of angular momentum, 160conservation of energy, 45

ballistic impacts, 352consistency

units of measurement, 7constants

gravitational constant, 7, 26continuous wave laser, 393convective heat transfer, 358conversion factors

SI/English System units, 6

428 ■I N D E X

coordinate axes rotationsrigid body motion, 75rotation matrices, 14

coordinate systems, 9–10Cartesian coordinate system, 9conversion between Cartesian and

spherical, 10origin, 10rotating by an angle, 151spherical coordinate system, 9, 10two-dimensional collision example, 153

coupled equations, 59crosswind

see wind effectscrowd behavior

Monte Carlo simulations, 411–418cryogenic propellants, 322cumulative distribution function

Gaussian distribution, 409probability functions, 407

curve fitting technique, 259CW (continuous wave) laser, 393

■Ddamping force, 38delta form

derivatives, 56density

air and water, 249buoyancy and, 250steel, 250

dependent variablederivatives, 16

derivatives, 15–18acceleration, 17, 18delta form, 56dependent variable, 16first-order derivative, 17independent variable, 16location and acceleration, 51rate of change of scalars, 17relationship of first/second-order

derivatives, 17second derivative, 51second-order derivative, 17velocity and acceleration, 50velocity and location, 50velocity derivative of z-component, 52

detonator explosive, 375differential

car racing simulations, 217differential equations, 18–19

boundary conditions, 59fourth order Runge-Kutta method, 56–58

programming, 59–63ordinary differential equations, 18

first-order, 56higher-order, 58–59solving, 55–69

partial differential equation, 18diode laser, 392direct drive

boat thrust, 250direct impact collisions, 160directional changes

turning airplanes, 296–297directional components

location, 51SpinProjectile class, 128velocity, 10, 11, 50

displacement, boats, 246buoyancy and density, 250

displacement hullboat hull types, 246residual drag, 257wave drag, 256

dopingsolid-state lasers, 392

draft, boats, 246drag coefficient

aerodynamic drag, 98, 99–103airplanes, 292baseball, 207bullets, 135car racing, 221flight, 307golf balls, 169Reynolds number, 99rockets, 335soccer ball, 191, 193surfing, 273three-dimensional shapes, 102two-dimensional shapes, 101

drag forceairplanes, 292–295

airplane efficiency factor, 293Cessna 172 Skyhawk, 295

429■I N D E X

drag equation, 294Flight Simulator, 307form drag, 292induced drag, 293lift over drag ratio, 295parasitic drag, 292pressure drag, 292skin friction drag, 292total drag equation, 294wing aspect ratio, 294

boats, 248, 254–258determining water contact area, 257form drag, 256skin friction drag, 255wave drag, 256

car racing simulations, 221Car class, 231

jet skis, 269projectile motion

see aerodynamic dragrockets, 331–332

Rocket Simulator, 338Saturn 1B velocity profiles, 333

DragProjectile classaerodynamic drag, 108–111baseball simulation, 204car racing simulations, 228getRightHandSide method, 109gravity and drag model, 108updateLocationAndVelocity method, 234

drive systemsboat thrust, 250–251jet ski simulations, 268

dye lasers, 392dynamic stability, airplanes, 315–316dynamite, 373

■Ee

use of e in scientific notation, 7effective power

boat thrust, 253Einstein, Albert, 385, 386elastic collisions, 142, 143electromagnetic spectrum

lasers, 391electrons, 386, 387elevator, airplanes, 277energy, 43–45

chemical energy, 43, 44conservation of energy, 45

ballistic impacts, 352friction, 44kinetic energy, 43–44mechanical energy, 43, 44potential energy, 43, 44SI units of, 43thermal energy, 43work-energy theorem, 43

engine brakingairplane thrust, 286car racing simulations, 226

engine torquecar racing simulations, 214, 216

engine typesjet engines, 291–292liquid-cryogenic, 321liquid-hypergolic, 322propeller engines, 285–290rockets, 321–322

F-1 rocket, 334nuclear rocket, 322RD-180 rocket, 334

solid-propellant engines, 322turbo engines, 291

final drive ratiospecific impulse for, 325

English System of Unitsacceleration, 23fundamental units of measure, 6linear momentum, 140mass, 23other units, 7SI unit conversion factors, 6units of measurement, 6

origins of, 7equal and opposite forces

Newton’s third law of motion, 24equations

angular momentum, 159coupled equations, 59gravitational force, 3indicating vectors in, 11scalars, 10trajectory of a projectile, 2use of Greek letters, 8vectors, 10

430 ■I N D E X

equations of motionacceleration equations, 85aerodynamic drag, 104–105basic concepts of projectile motion, 84constant gravitational force, 53force equations, 85gravity-only projectile model, 97wind effects on projectile motion, 115

error functionsolving heat conduction equation, 362

getErrorFunction method, GasTank class, 366

escape velocity, orbits, 343Euler angle

rotation matrices, 14Euler’s method, 57excimer laser, 391explosions, 371–383

basic principles, 372–373blast damage, 375–380

TNT equivalence of bombs, 379–380chemical explosions, 371explosion models, 379

TNT explosion model, 376fragmentation devices, 380–381nuclear explosions, 371, 381–382references for further reading, 383terminology, 372types of explosive, 373–375

explosivesdetonator explosive, 375types, 373–375

exponential distribution function, 410

■FF-1 rocket engine, 334final drive ratio

car racing simulations, 217fins

vertical fin, 277first law of motion, Newton’s

see under Newton’s laws of motionfirst-order derivative, 17

ordinary differential equation, 19second-order derivative relationship, 17

first-order ordinary differential equationlocation and velocity, 56spring motion example, 63

fission, 371

flaps, airplanes, 277, 283Flight Simulator, 302

flash pointexplosion terminology, 372

Flight Simulatorairplanes, 301–310

GUI display, 303classes implementing, 302

FlightSimulator class, 309fluid dynamics

evaluating airfoil lift, 280flying

see airplanesfootball simulation, 208force

acceleration and, 50English System unit of, 23impulsive force, 141linear impulse of force, 140Newton’s second law of motion, 23SI unit of, 23summarized, 46two-body linear collisions, 144

force diagramairplanes, 278boats, 247–248box, 40box on ramp, 41car racing simulations, 212golf ball in flight, 176gravity and drag effects, 105gravity-only model, 85gravity, drag, and Magnus force, 125rockets, 323sailing, 270sliding bowling ball, 78

force equationsaerodynamic drag, 105–106gravity-only model, 85

force of gravityweight, 24

force vector, 24–25forces

centripetal force, 38–39force balances, 39friction, 31–38gravitational force, 25–31Newton’s third law of motion, 24normal force, 32

431■I N D E X

springs, 37–38types of, 25–39variable magnitude force, 33

form dragairplane drag, 292hydrodynamic drag, 256

Fountain Lightningmodeling boat acceleration and

velocity, 259acceleration profile, 260

speedboat simulations, 261FountainLightning class

getRightHandSide method, 264Powerboat class, 263

Fourier’s law, 359fourth order Runge-Kutta method

fundamentals of, 58initial value problem, 59solving differential equations, 56, 58

programming for, 59–63spring motion example, 63–67

solving higher-order ODEs, 58fragmentation devices

explosions, 380–381geometrical constants, 380

frames of reference, 9–10basic concepts of projectile motion, 84inertial frame of reference, 10noninertial frame of reference, 10

FreeKick classactionPerformed method, 198soccer free kick game, 196

FreeThrow classbaseball simulation, 204

frequency of rotationrotational motion, 70

friction, 31–38aerodynamic drag, 98coefficient of friction, 32, 34collisions with, 160–164energy, 44force balances, 40golf club-ball impact, 172thermal energy, 45work overcoming, 41

frictional forcesee coefficient of friction

frictional impulsetwo-dimensional oblique collisions, 163

Fronsperger, Leonhart, 319Froude, William, 256full-body aerodynamics

see aerodynamics, airplanesfundamental units of measure, 6fuselage, airplanes, 276fusion, 371

■Ggame simulations

see simulationsgames programming

physics and, 4GameUpdater class, 29, 30, 36

actionPerformed method, 68, 95GasTankSimulator class, 368LaserSimulator class, 399PaddleGame class, 155SoldierGame class, 416SphereCollision class, 148

gas lasers, 391–392excimer laser, 391radiation types, 392

Gas Tank SimulatorGasTank class, 365GUI display, 364heat conduction, 364–369references for further reading, 370solving heat conduction equation, 365

GasTank classGas Tank Simulator, 365getErrorFunction method, 366get/set methods, 365getTemperature method, 366

GasTankSimulator class, 367actionPerformed method, 367GameUpdater class, 368Timer object, 367

Gaussian distributionprobability functions, 408–410Soldier Game, 412, 416

gears, 217, 219geostationary orbit, 343geosynchronous orbit, 343getErrorFunction method

GasTank class, 366get/set methods

GasTank class, 365SimpleRocket class, 337

432 ■I N D E X

getLocationAndVelocity methodPowerboat class, 262

getRightHandSide methodCar class, 231DragProjectile class, 109FountainLightning class, 264GolfBall class, 180ODE class, 61, 62Powerboat class, 263PropPlane class, 305SimpleProjectile class, 91SimpleRocket class, 338, 339SoccerBall class, 196SpinProjectile class, 127SpringODE class, 65WindProjectile class, 117

getTemperature methodGasTank class, 366

Goddard, Robert, 320golf

brief history of, 167Golf Game

aerodynamic drag, 111–113golf simulation, 178–184gravity-only model, 91–96SimpleProjectile class, 91–96wind effects on projectile motion, 119–121

golf simulation, 167–189equipment specifications, 168–170Golf Game, 178–184Magnus force for spinning golf ball, 176–178modeling ball in flight, 176–178modeling club-ball impact, 170–175

collision analysis, 174–175friction effects, 172–174

putting, 184–189computing distance ball travels,

188–189impact phase, 185rolling phase, 187–188skid phase, 185–187slope effects, 189

references for further reading, 209rolling friction, 187

GolfBall class, 178, 179getRightHandSide method, 180

GolfGame classactionPerformed method, 181GameUpdater class, 95

GolfGame2 classactionPerformed method, 112aerodynamic drag, 111–113

GolfGame3 classactionPerformed method, 120wind effects on projectile motion, 119–121

GolfGame4 classactionPerformed method, 131spin effects on projectile motion, 129–132

gravitational accelerationaltitude effects, 330at Earth’s surface, 52force, mass, and, 52Rocket Simulator, 339

gravitational force, 25–31acceleration due to, 26equation for, 3equations of motion for constant, 53generating moments, 310gravitational constant, 26Gravity Game, 27rockets, 330–331rolling motion, 76satellites, 342summarized, 46use of e in scientific notation, 7

gravitational potential energy, 44gravity

escape velocity, orbits, 343forces acting on boats, 248modeling gravitational force, 3

Gravity Game, 27, 29gravity-only model

projectile motion, 84–97equations of motion, 97force and acceleration equations, 85–86Golf Game, 91–96location equations, 86–88SimpleProjectile class, 88–91summary, 97time to reach trajectory apex, 88

Greek letters, 8–9ground effect

aerodynamics, airplanes, 300Flight Simulator, 307

ground speedaerodynamics, airplanes, 300

ground state, 387guidance systems, 347

433■I N D E X

gunpowder, 373guns

projectile motion of bullets, 134–135gyroscopic precession

motorcycle racing simulations, 241

■Hhand grenade

modeling fragmentation devices, 380headwind

see wind effectsheat conduction, 358–369

Fourier’s law, 359Gas Tank Simulator, 364–369heat conduction equation, 360–361

1-D heat conduction equation, 360, 361Gas Tank Simulator, 365solving, 361–363specific heat capacity, 360temperature derivative, 360thermal diffusivity, 361thermal diffusivity values, 361

thermal conductivity, 359heat energy, 358heat of explosion, 372helium-neon laser, 390Hellfire missile, 348High Energy Laser Experimental (HELEX), 395higher-order ODEs

solving differential equations, 58–59high-speed turns

car racing simulations, 238hockey simulation, 209Hooke’s Law, 37horizontal stabilizer, airplanes, 277hulls, boats, 246

boat hull types, 246–247displacement hull, 246hull efficiency, 254hull speed, 257planing hull, 247

hybrid rocket engine, 322hydrodynamic drag

boat hull types, 247determining water contact area, 257drag force, boats, 254forces acting on boats, 254–257form drag, 256other drag components, 257

skin friction drag, 255wave drag, 256

■Iimmovable objects

collisions with, 146impacts

see ballistic impactsimpeller

jet ski drives, 268impulsive force, 141

linear impulse of force, 140inboard drive system, 250Inboard/Outboard (I/O) drive system, 251incidence angle

airfoils, airplanes, 279independent variable

derivatives, 16induced drag

airplane drag, 293inelastic collisions, 142–143inertia

angular momentum and, 159moment of inertia, 72Newton’s first law of motion, 22

inertial frame of reference, 10initial value problem, 59insulating materials, 359integration, 52International System of Units

see SI unitsinverse cumulative distribution function, 407

Gaussian distribution, 409

■JJavan, Ali, 386jet drive

boat thrust, 251jet ski simulations, 268

jet enginesairplane thrust, 291–292turbo engines, 291

jet ski simulations, 268–269see also boatsdrive systems, 268Kawasaki 800 SX-R, 269thrust and drag, 269

jib, boats, 269Joule, SI units, 42

434 ■I N D E X

■KKawasaki 800 SX-R, 269keel, boats, 269kinematics, 49–80

see also projectile motionbowling ball kinematics, 77

kinetic coefficient of friction, 34kinetic energy, 43–44

elastic and inelastic collisions, 142knot, boats, 246Kutta, Martin, 56

■Llaminar flow

aerodynamic drag, 103–104Reynolds number, 104soccer ball in flight, 191

land minesmodeling fragmentation devices, 380

landing airplanesaerodynamics, 301flaps, 283Flight Simulator, 302

landing gear, airplanes, 277lasers, 385–402

absorption factor, 395acronym (LASER), 385continuous wave (CW) laser, 393creating laser systems, 400–401damage caused by, 395–397diode laser, 392dye lasers, 392electromagnetic spectrum, 391gas lasers, 391–392helium-neon laser, 390history of, 385–386how lasers work, 389–391introduction to atoms, 386–389laser beam generation, 390Laser Simulation, 397–400military lasers, 393–395photons, 385, 389pulse laser, 393pumping the laser, 389Rayleigh/Raman scattering, 401references for further reading, 402semiconductor laser, 392solid-state lasers, 392types of lasers, 391–393

visual effects, 401wavelength of emitted radiation, 391

LaserSimulator classactionPerformed method, 398GameUpdater class, 399Laser Simulation, 397Timer object, 398

laws of motionsee Newton’s laws of motion

leading edgeairfoils, airplanes, 278

least squares approximation technique, 259length, unit of, 6letters

use of Greek letters, 8lift coefficient

aerodynamics, airplanes, 300evaluating airfoil lift, 281flaps, airplanes, 283Flight Simulator, 303, 306, 307NACA 0012 and 2412 airfoils, 281spin effects on projectile motion, 124

lift forceairplanes, 278–285

airfoils, 278, 279, 280center of pressure, 284generating moments, 310lift over drag ratio, 295

rockets, 333Rocket Simulator, 336, 339

line of actioncollisions, 144

linear collisionstwo-body linear collisions, 143–149

coefficient of restitution, 145linear impulse of force, 140linear momentum, 140–141

conservation of, 141–142equation for, 140system of objects, 141

liquid-cryogenic engines, 321liquid-hypergolic engines, 322liquid-propellant rocket, 321location

acceleration and, 51equations of motion, 53gravity-only model, 86vectors, 51velocity and, 50, 56

435■I N D E X

location equationsaerodynamic drag, 107

■MMagnus effect/force

effects of spin on flight of baseball, 207golf ball in flight, 176–178Golf Game, 184soccer ball in flight, 194spin effects on projectile motion, 123–125

directional components, 125Maiman, Theodore, 386mainsail, boats, 269manual transmission

car racing simulations, 220maser, 386mass

center of mass, 73–74moment of inertia, 72Newton’s second law of motion, 23units of, 6, 23weight and, 23

mass ratio, 344mast, boats, 269matrices, 12–15

multiplying matrices, 13rotation matrices, 13

McCauley 7557 propellerpropeller efficiency coefficient, 286

meanprobability functions, 406

measurementsunits of measurement, 6–7

mechanical energy, 43, 44mechanics, Newtonian, 21–46military lasers, 393–395

Airborne Laser, 393–394High Energy Laser Experimental, 395Space Based Laser, 394

missiles, 347–349air-to-air missiles, 347air-to-surface missiles, 347missile guidance, 347missile specifications, 348Sidewinder air-to-air missile, 347surface-to-air missiles, 347

modeling projectile motionsee projectile motion

modeling motion of objectssee derivatives

moment armtorque, 71

moment of inertiaangular acceleration, 72for common objects, 73powerboat turns, 267two-dimensional oblique collisions, 164

moments, airplanes, 310–311moment coefficient data, 311

momentumangular momentum, 159–160ballistic impacts, 356–357linear momentum, 140–141

conservation of, 141–142linear impulse of force, 140

thrust, rockets, 324Monte Carlo simulations, 410–421

computing value of Pi, 418crowd behavior, 411–418history of, 411

motionsee also projectile motionequations of motion, 53rigid body motion, 73–79rolling motion, 75rotational motion, 69–73translational equations of motion, 51–69translational motion, 49–51

motion of objects, modelingsee derivatives

motion of projectilessee projectile motion

motorcycle racing simulations, 240–242car racing compared, 240gyroscopic precession, 241turning a motorcycle, 241

multiplicationmultiplying matrices, 13

multistage rockets, 345–346Saturn V rocket, 345

stage data for, 346

■NNACA 0012 and 2412 airfoils, 281

Flight Simulator, 306lift coefficient data, 281moment coefficient data, 311

436 ■I N D E X

nautical terminologysee under boats

neodymium-Yag laser, 392neutrons, 386Newton, Isaac, 22Newton’s laws of motion

basic concepts of projectile motion, 84first law of motion, 22–23

gravitational force, 25second law of motion, 23–24

angular acceleration, 72center of mass, 73conservation of linear momentum, 141creating lift, 279force, acceleration, velocity, and

location, 50gravitational force equation, 26gravity-only model, 85rocket thrust, 324rolling friction, 187torque, 71translational acceleration, 84translational equations of motion, 51translational motion, 49velocity derivative equation, 140work-energy theorem, 43

third law of motion, 24creating lift, 280friction and normal force, 32linear impulses, 141

Newtonian mechanics, 21–46NIJ body armor classification standards, 355

bullets used in tests, 355references for further reading, 370

nitroglycerinetypes of explosives, 373

Nobel, Alfred, 373noninertial frame of reference, 10nonlinear probability function, 406normal distribution

see Gaussian distributionnormal force

friction and, 32nose, 321notation

Greek letters, 8scientific notation, 7–8summation notation, 8

nuclear explosions, 371, 381–382

nuclear rocket engines, 322nucleus (of atoms), 386

■Oobjects, modeling motion of

see derivativesoblique collisions, 160

three-dimensional collisions, 164two-dimensional collisions, 162

ODE classgravity-only model

SimpleProjectile class, 89programming fourth order Runge-Kutta

method, 60, 61ODESolver class

programming fourth order Runge-Kutta method, 61

orbits, 342–345circular orbits, 343delivering payloads into, 344Earth’s rotation, 344escape velocity, 343geostationary orbit, 343geosynchronous orbit, 343polar orbit, 343satellites, 342

ordinary differential equationmotion of spring as function of time, 19

ordinary differential equation (ODE), 18origin, coordinate systems, 10outboard drive

boat thrust, 251overpressure

blast damage, 375explosion terminology, 372, 373personal injury due to, 376structural damage due to, 375TNT explosion model, 377, 378

■Ppaddle game

two-dimensional collisions, 153–157PaddleGame class, 154

actionPerformed method, 155parabola

gravity-only model, 87parasitic drag

airplane drag, 292partial differential equation (PDE), 18

437■I N D E X

payloadsdelivering into orbit, 344rocket terminology, 321

performanceeffect of using physics, 3

performance datamodeling acceleration/velocity for

boats, 258photons, 387, 388, 389

description, 385lasers, 389wavelength of, 389

physics, reasons to useachieving realism, 2avoiding unrealistic looking games, 1game performance, 3improved programming, 4

Picomputing value of, 418

PiEstimator class, 419pitch, airplanes, 277

aerodynamics, 297airplane stability, 313pitch angle, 296propellers, 252

pitch, rocketsRocket Simulator, 335

Planck constant, 389planes

see airplanesplaning hull

boat hull types, 247determining water contact area, 257

plastic explosivestypes of explosives, 374

point of explosion, 372Poisson distribution function, 410polar orbit, 343Porsche Boxster S

car racing simulations, 215gear ratios, 218torque curve for, 222

port, boats, 246potential energy, 43, 44

gravitational potential energy, 44power

car racing simulations, 216SI unit of, 45work and, 45

power curvecar racing simulations, 216

power drop-off factoraltitude effects on propeller engine, 288

Powerboat classFountainLightning class, 263getLocationAndVelocity method, 262getRightHandSide method, 263speedboat simulations, 261

powerboat simulationssee speedboat simulations

pressure and pressure ratioaltitude effects, rockets, 326

pressure dragaerodynamic drag, 98airplane drag, 292

pressure thrust, rockets, 324probability functions, 405–410

crowd behavior, 411–418cumulative distribution function, 407Gaussian distribution, 408–410Monte Carlo simulations, 410–421nonlinear probability function, 406standard deviation, 408

programmingphysics and games programming, 4

programming languagesscientific notation, 8

projectile motion, 83–137aerodynamic drag, 98–114arrows, 136–137ballistic impacts, 351–358basic concepts, 84bullets, 134–135cannonballs, 135–136gravity-only model, 84–97

force and acceleration equations, 85–86Golf Game, 91–96location equations, 86–88SimpleProjectile class, 88–91time to reach trajectory apex, 88

momentum in ballistic impacts, 356spin effects, 122–133wind effects, 114–122

propellant massRocket Simulator, 340

438 ■I N D E X

propellersairplane thrust, 285–290

altitude effects, 288Cessna 172 Skyhawk, 289engine brake power, 286power drop-off factor, 288propeller advance ratio, 286propeller efficiency coefficient, 286

boat thrust, 251, 253pitch, 252propeller advance ratio, 286propeller efficiency coefficient, 285, 286

PropPlane classFlight Simulator, 302

get/set methods, 305getRightHandSide method, 305

propulsive efficiency, coefficient ofboat thrust, 253

protons, 386, 387pulse laser, 393putting

golf simulation, 184–189

■Qquadratic equation

gravity-only model, 86quarter chord point, 310

■Rradians

angular velocity, 70radiative heat transfer, 358Random class, 404random number generation, 404–405

getting different sequences of, 404Rayleigh/Raman scattering, 401RD-180 rocket engine, 334real slip ratio

propellers, 252resetDisplay method

BoatSimulator class, 266residual drag, 257resistance, coefficient of

drag force, boats, 254restitution, coefficient of, 145Reynolds number

baseball, 207drag coefficient, 99drag coefficient, bullets, 135drag coefficient, cannonballs, 135

laminar flow, 104skin friction drag, 255soccer ball in flight, 192

Rhodamine 6G dye laser, 392rigid body motion, 73–79

bowling ball kinematics, 77center of mass, 73–74coordinate axes, 75rolling motion, 75

RK4Spring class, 66Robin’s effect

spin effects on projectile motion, 123rocket equation, 324rockets

drag, 331–332engine types, 321–322

futuristic engine types, 322hybrid engine, 322liquid-cryogenic engines, 321liquid-hypergolic engines, 322nuclear engines, 322solid-propellant engines, 322specific impulse for, 325

force diagram, 323gravity, 330–331historical background, 319–320lift, 333missiles, 347–349multistage rockets, 345–346orbits, 342–345references for further reading, 349Rocket Simulator, 334–342

classes implementing, 336GUI display, 334, 335, 339

stability, 333terminology, 321

mass ratio, 344thrust, 323–330

altitude effects, 326calculating atmospheric

measurements, 327Newton’s second law, 324specific impulse, 325

wind effects, 333RocketSimulator class

actionPerformed method, 340GUI display, rocket simulator, 339

roll, airplanes, 277aerodynamics, 297

439■I N D E X

rolling frictionCar class, 231car racing simulations, 221golf simulation, 187

rolling motion, 75rotation matrices, 13

coordinate axes rotations, 14Euler angle, 14

rotational accelerationbasic concepts of projectile motion, 84rolling motion, 77

rotational axisfor common objects, 73

rotational motion, 69–73angular acceleration, 70angular velocity, 70axis of rotation, 71frequency of rotation, 70rigid body motion and, 73rolling motion, 75, 76spherical coordinate system, 10tangential velocity, 71torque, 71

rotational spin ratiospin effects on projectile motion, 124

rotational velocityrolling motion, 77

ruby laser, 392rudder, airplanes, 277Ruggieri, Claude, 320Runge, Carl, 56Runge-Kutta method, fourth order

see fourth order Runge-Kutta methodrungeKutta4 method

ODESolver class, 61

■Ssailing

see also boatscenter of mass of boat, 270physics of sailing, 269–270stability of boat, 270terminology, 269thrust, 270

satellitescentripetal force, 342gravitational force, 342orbits, 342

Saturn 1B velocity profiles, 333Saturn V rocket, 345

stage data for, 346scalars, 10–12

rate of change of, 17Schawlow, Arthur, 386scientific notation, 7–8

programming languages, 8use of e, 7

second derivatives, 51second law of motion, Newton’s

see under Newton’s laws of motionsecond-order derivative, 17

first-order derivative relationship, 17ordinary differential equation, 19

semiconductor laser, 392semi-displacement hull, 247setGearRatio method

car racing simulations, 233setSpeed method

Soldier class, 417shaft transmission efficiency

boat thrust, 253shiftGear method

car racing simulations, 230ships

see boatsshooting

two-point boundary problem, 59Shuffleboard Game, 34SI units

acceleration, 23energy, 43English System of Units and, 6force, 23fundamental units of measure, 6Joule, 42linear momentum, 140mass, 23power, 45units of measurement, 6work, 42

Sidewinder air-to-air missile, 347missile specifications, 348

SimpleProjectile classGolf Game, 91–96gravity and drag model

DragProjectile class, 108

440 ■I N D E X

gravity-only model, 88–91getRightHandSide method, 91ODE class, 89

updateLocationAndVelocity method, 90, 95SimpleRocket class

get/set methods, 337getRightHandSide method, 338, 339Rocket Simulator, 336

simulationsbaseball, 204–207basketball, 199–204car racing, 212–240crowd behavior, 411–418flight, 301–310football, 208Gas Tank Simulator, 364–369golf, 167–189hockey, 209jet skis, 268–269lasers, 397, 400linear collisions, 146Monte Carlo simulations, 410–421motorcycle racing, 240–242rockets, 334–342soccer, 189–199speedboats, 261–267tennis, 209vertical ascents, 342

skidpad testcar racing simulations, 236

skin friction dragairplane drag, 292hydrodynamic drag, 255skin friction coefficient, 255viscosity effect, 255

Sobrero, Ascanio, 373soccer

brief history of, 189soccer simulation, 189–199

equipment specifications, 190free kick game, 195–199modeling ball in flight, 191–194

laminar and turbulent drag, 191Magnus force, 194

modeling ball-foot impact, 191SoccerBall class, 196Soldier class, 414

setSpeed method, 417

Soldier Game, 412SoldierGame class, 415

actionPerformed method, 416GameUpdater class, 416Timer object, 415

solid objectsballistic impacts, 351–358heat conduction, 358–369

solid-propellant engines, 322solid-propellant rocket, 321solid-state lasers, 392Space Based Laser (SBL), 394

time to destroy a missile, 397span, airplanes, 277specific heat capacity, 360specific impulse, rockets, 325speedboat simulations, 261–267

see also boatsaerodynamic drag, 258boat hull types, 247Boat Simulator, 261boat thrust, stern drive, 251Fountain Lightning, 261Powerboat class, 261powerboat turns, 267

SphereCollision classactionPerformed method, 148linear collision simulator, 146

spherical coordinate system, 9, 10conversion to Cartesian system, 10

spin effectsoblique collisions, 161projectile motion, 122–133

tilting spin axis, 132SpinProjectile class

getRightHandSide method, 127golf simulation, 178, 179soccer free kick game, 196spin effects on projectile motion, 125–129

directional components, 128GolfGame4, 129

updateLocationAndPosition method, 127sport simulations

see simulationsspring force, 46Spring Simulator, 67SpringODE class

spring motion example, 63–66

441■I N D E X

springs, 37–38forces, 37, 38Hooke’s Law, 37motion of spring as function of time, 19solving higher-order ODEs, 58spring motion example, 63–67spring motion simulator, 67–69

stability, airplanes, 313–314dynamic stability, 315–316pitch stability, 313stability and trim, 314–315static stability, 315

stability, rockets, 333stabilizers

horizontal stabilizer, 277vertical stabilizer, 277

stall, airplanes, 282standard deviation

probability functions, 408Soldier Game, 417

starboard, boats, 246state variables

energy, 43static coefficient of friction, 34static stability, airplanes, 315steel

ballistic impacts on steel armor, 353density of steel, 250

stern, boats, 246stern drive, 251summation notation, 8surface drives, 251surface-to-air missiles, 347surfing, 270–273

buoyancy and balance, 271catching a wave, 272center of mass, 271drag coefficient, 273physics of waves, 272turning, 272

system of objects, 141systems of units, 6–7

consistency in use of, 7

■Ttail, airplanes, 276tailwind

see wind effects

takeoff, airplanesaerodynamics, 299flaps, 283Flight Simulator, 302

tangential velocityrotational motion, 71

temperature derivativeheat conduction equation, 360

temperature units, 6tennis simulation, 209terminal velocity

aerodynamic drag, 108thermal conductivity, 359thermal diffusivity, 361thermal energy, 43

friction, 45heat energy, 358

thin airfoil theory, 280third law of motion, Newton’s

see under Newton’s laws of motionThompson “F-Formula”

armor penetration, 353three-dimensional collisions, 157

oblique collisions, 164three-dimensional space

coordinate systems, 9thrust force

airplanes, 278, 285–292jet engines, 291–292propeller engines, 285–290

boats, 248, 250–254boat drive systems, 250propellers, 251

jet skis, 269rockets, 323–330

altitude effects, 326calculating atmospheric

measurements, 327momentum thrust, 324pressure thrust, 324Rocket Simulator, 335, 338specific impulse, 325

sailing, 270vector, powerboat turns, 267

time, units of, 6

442 ■I N D E X

Timer objectactionPerformed method, 121BeanBag class, 54car racing simulations, 234GasTankSimulator class, 367Golf Game, 91, 95Gravity Game, 29, 30LaserSimulator class, 398Shuffleboard Game, 35SoldierGame class, 415speedboat simulations, 261SphereCollision class, 148SpringSimulator class, 68

TNT (trinitrotoluene), 374TNT equivalence model

blast damage, 378TNT explosion model

blast damage, 376–378torque

angular acceleration, 78angular impulse of torque, 159angular momentum, 159boat thrust, 250bowling ball kinematics, 78Car class, 231car racing simulations, 218, 222moment arm, 71rotational motion, 71

total drag equation, airplanes, 294Townes, Charles, 386traction force

car racing simulations, 236trailing edge

airfoils, airplanes, 278trajectory, projectiles

equation for, 2gravity-only model, 87

translational accelerationbasic concepts of projectile motion, 84rolling motion, 76, 77

translational equations of motion, 51–69Beanbag Game, 53

translational motion, 49–51force, acceleration, velocity, and

location, 50Newton’s second law, 49rigid body motion and, 73rolling motion, 75, 76

translational velocitybasic concepts of projectile motion, 84bowling ball kinematics, 79rolling motion, 76, 77

transmissionscar racing simulations, 217

transom, boats, 246trim, airplanes, 311–312

stability and trim, 314–315Tsiolkovskii, Konstantin, 320turbo jet engines, 291turbulent flow

aerodynamic drag, 103–104soccer ball in flight, 191

turningaerodynamics, airplanes, 296–297

two-dimensional collisions, 149–157example, 152oblique collisions, 162paddle game, 153–157

two-dimensional spacecoordinate systems, 9

two-point boundary problem, 59

■Uunit vector, 11units of measurement

consistency in use of, 7systems of units, 6–7

updateConditions methodUSatm76 class, 327

updateDisplay methodGravityGame class, 36

updateLocationAndPosition methodSpinProjectile class, 127

updateLocationAndVelocity methodDragProjectile class, 234SimpleProjectile class, 90, 95

gravity and drag model, 109WindProjectile class, 117

updatePositionAndVelocity methodSpringODE class, 65SpringSimulator class, 69

USatm76 classcalculating atmospheric measurements,

327, 338Rocket Simulator, 336updateConditions method, 327

443■I N D E X

■Vvariable magnitude force, 33vector cross product, 12vectors, 10–12

computing magnitude of vector, 11force vector, 24indicating in an equation, 11indicating that terms are, 50location, 51unit vector, 11vector cross product, 12velocity, 50

Vee drivesboat thrust, 251

velocitysee also translational velocityacceleration and, 50acceleration data for Kawasaki 800 SX-R, 269airplane stall speed, 282angular velocity, 70apparent velocity, 114car racing simulations, 222

maximum velocity, 224computing magnitude of vector, 11directional components, 10, 11, 50equations of motion, 53escape velocity, orbits, 343gravity-only model, 86location and, 50, 56modeling for boats, 258–261

acceleration data, 259tangential velocity, 71terminal velocity, 108vectors, 50

velocity equationsaerodynamic drag, 106–107angular momentum, 159post-collision velocity equations, 144two-dimensional collisions, 150

example, 152oblique collisions, 163

vertical ascent simulationRocket Simulator, 342

vertical directiongravitational force, 52velocity derivative, 52

vertical fin, airplanes, 277vertical stabilizer, airplanes, 277

viscosityskin friction drag, 255water, 255

■Wwake fraction, 253

efficiency coefficients, 254wake velocity, propellers, 252water

density of air and water, 249viscosity, 255

WattSI units, 45

wave dragcalculating, 257hydrodynamic drag, 256wave velocity, 256

waves, 272weight

mass and, 23wheel radius

car racing simulations, 219wheel torque

car racing simulations, 217wheel traction

car racing simulations, 236Wilbrand, Joseph, 374wind effects

projectile motion, 114–122Rocket Simulator, 336rockets, 333

WindProjectile classgetRightHandSide method, 117updateLocationAndVelocity method, 117wind effects on projectile motion, 116–118

wing aspect ratioairplane drag, 294

wings, airplanes, 276see also airfoils, airplanesairplane trim, 311

work, 41–42movement and work, 41power and, 45SI units of, 42work-energy theorem, 43

Wright brothers, 2761903 Wright Flyer, 276

444 ■I N D E X

■Yyaw, airplanes, 277

aerodynamics, 297yaw angle

projectile motion of bullets, 134

■Zz-direction, 52