Analysis of Golf Ball MotionBrad PeirsonEGR 312 – DynamicsInstructor: Prof. WaldronSchool of EngineeringPadnos College of Engineering and ComputingGrand Valley State UniversityMarch 23, 2007AbstractProjectile motion is a problem addressed in engineering, dynamics and physicscourses as well as in real world applications, namely ballistics. There are many potentialmethods for studying the path of a projectile; four of the most common methods areaddressed in this report. The four methods used in analyzing the flight of a golfballwere analysis neglecting air friction, analysis assuming constant air friction and twoanalyses using variable air friction: central differences and Runge-Kutta integration.The results of the flight path of the golf ball were plotted on the same set of axesfor comparison. There are obvious differences between the four methods with respectto the predicted path of the golf ball, however because of the way the Runge-Kuttasolution is calculated this metho d produces the most accurate results.1 IntroductionThe purpose of this report is to compare four different methods of analyzing the flight path ofa golf ball. This type of dynamics problem is considered simple projectile motion. Often, inorder to simplify the calculations, the effects of air resistance on the projectile are neglected.Neglecting such effects are adequate for simple analyses in order to obtain an approxi-mate trajectory, however there are high precision applications that require such affects beaccounted for. Because actual projectiles experience air resistance in the form a drag force,an accurate model of projec tile motion must include drag. Including drag forces in the anal-ysis of motion produces a series of differential equations that represent the motion. Thedifferential equations that represent projectile motion are:dxdt= u (1)dydt= v (2)dudt= −CDRe24τu (3)dvdt= −CDRe24τv − g (4)where u is the velocity of the projectile in the x-direction, v is the velocity of the projectilein the y-direction, CDis the drag coefficient, Re is the Reynold’s number and τ is the timeconstant.The system defined by equations 1 through 4 can also be used to analyze the motion ifdrag is neglected. In this case the first terms in equations 3 and 4 will be equal to zero.Using the above system of equations a comparison of the motion both accounting for andneglecting drag can be obtained.122.1 Derivation of Differential EquationsThe first step in the comparison of four analysis methods is to understand how the differentialequations 1 through 4 were derived. The following relationships were given in the projectdescription and are necessary for this derivation:D =12CDρAV2(5)Re =ρV Dµ(6)τ =m3πdµ(7)where D is the drag force, ρ is the air density, A is the cross sectional area of the projectileat it’s widest point, V is the velocity of the projectile, d is the diameter of the projectile atits widest point and m is the mass of the projectile.The differential equations can b e derived by applying Newton’s second law of motion tothe free body diagram and kinetic diagram given in Figure 1.Figure 1: Free Body and Kinetic Diagrams for the Golf Ball in Flight2.2 Differential Equations in X-DirectionThe derivation of the differential equation pertaining to the x-direction is as follows:XFextx= max−D = max2From equation 1 we see thatax=dudtUsing the substituting the above results into the force balance equation gives−12CDρAV2= mdudt−CDρAu22m=dudtThe cross sectional area of the golf ball, A, is a circle.A =πd24−CDρπd2u28m=dudtManipulating the given statement for the time constant, τ , gives13τµ=πdm−CDρdu224τµ=dudtAnd finally, substituting the statement for the Reynold’s Number into the equation givesthe final differential equation for velocity in the x-direction.Re =ρduµdudt= −CDRe24τu2.3 Differential Equations in Y-DirectionThe derivation of the equations in the y-direction are identical to those in the x-direction,except that the force of gravity, mg, acts in the y-direction.−D − mg =Xmayay=dvdt3−D − mgm=dvdt−CDρAv22m− g =dvdtA =πd24−CDρπd2v28m− g =dvdt13τµ=πdm−CDρdv224τµ− g =dvdtRe =ρdvµdvdt= −CDRe24τv − g2.4 Analysis of MotionThe following information was given for the flight of the golf ball:Weight (mg) 1.5ozDiameter (d) 1.75inAir Viscosity (µ) 0.375 × 10−6lb−secftInitial Velocity (V0) 120ftsecInitial Angle (θ) 30◦from horizontalAcceleration due to gravity (g) 32.2ftsec242.4.1 Neglecting FrictionThe simplest method of analyzing the flight of the golf ball is to neglect the effects of thedrag force. This was the first method used in the comparison of analysis techniques.Assuming no air friction on the golf ball, the accelerations in the x and y-directionsbecome:ax= 0ay= −gFrom kinematics, the equation for distance, x, as a function of time isx(t) = x0+ V0xtwhere x0is the projectile’s initial position and V0xis the initial velocity in the x-direction.V0xcan be expressed in terms of the initial velocity, V0, and the initial launch angle, θ:V0x= V0cosθV0x= 103.92ftsecWhen the initial position of the golf ball is assumed to be located at the origin, (0, 0),the distance of the ball as a function of time becomes:x(t) = 0 + (V0cosθ)tThe equation for the height, y, of the ball in terms of time is:y(t) = y0+ V0yt −12gt2where y0is the initial height and V0yis the initial velocity in the y-direction. Again, V0ycan be expressed in terms if the initial velocity, V0, and the launch angle, θ.V0y= V0sinθV0y= 60ftsecIf the starting position is again assumed to be the origin, the height of the ball withrespect to time is given as:y(t) = 0 + (V0sinθ)t −12gt2Using equations from kinematics and projectile motion, the equation f or the maximumrange of the golf ball is5xmax=V20sin2θgxmax= 387.29ftAnd the equation for the maximum height of the golf ball isymax=(V0sinθ)22gymax= 55.90ftThe equations for the velocities in the x and y-directions can be found by taking the timederivatives of the equation for distance, x, and height, y:dxdt= udydt= vu(t) = V0cosθv(t) = V0sinθ − gtThese position equations were plotted in Figure 2 for comparison with the other analysismethods.2.4.2 Constant Air FrictionThe analysis with constant drag force will produce more accurate results that the analysiswith no drag. The differential equations, equations 3 and 4, were used in the analysis withconstant drag.To simplify the math in this model, the time