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Simulation of a Golf Ball Trajectory

Winter 2004

 

Contents

Introduction

Computer simulation can be used to model a variety of real life problems.  This project was prepared to realistically predict the trajectory followed by a golf ball travelling through still air. The golf ballís motion is determined by solving an initial value problem of first and second order non-linear ordinary differential equations, which contained relations to describe lift and drag forces produced by air flow over the spinning golf ball. Using these and other principle relations, the ordinary differential equations were produced and a solution was determined by the most efficient numerical integration technique approach. It is noted in the project to neglect the Magnus effect and assume the motion of the golf ball to occur on the x-z plane. The relations used to arrive at the non-linear ordinary differential equations consisted of relationships that describe lift force (FL), drag force (FD), drag coefficient (CD), lift coefficient (CL), velocity (V), horizontal range (x), height (z),  and velocity argument (").

Golf Ball Physics

Initial velocity, spin and angle of flight are the three main elements that determine the range and the trajectory of the golf ball.

 Initial velocity

This refers to the velocity of the ball leaving the club head at the instant of impact.  At the moment of impact the ball deforms.  The energy from this deformation is stored within the ball.  As deformation reaches its peak, the ball restores its self to its original shape.   The power of this restitution is what causes the ball to fly off the club face.  Although initial velocity depends primarily on head speed, the material used to manufacture the club head, point of contact, swing path, the balls restitution, can all affect the initial velocity.  Figure 1 shows the correlation between head speed and initial velocity.

  

Figure 1 Correlation between head speed and initial velocity

 Initial angle of flight

This refers to the angle at which the ball leaves the club head upon impact.  Initial angle is determined by the angle of incidence, the degree of ball deformation, and the friction generated between the club and the ball [1].is refers to the velocity of the ball leaving the club

Spin

This refers to the balls revolutions.  Spin (or back spin) is determined by the head speed, swing path, and the deformation and material of the ball and the club.  The lift created by the balls spin causes the ball to travel along high trajectories.  Poorly hit balls tend to spin sideways, which becomes the causes of hooks and slices.

A forward moving golf ball spins in the direction opposite to its target.  The back spin generates power the lifts the ball higher.  Greater distance is achieved if spin is minimal.  Back spin is used by professional golfers to slow a ball as it reaches its target. 

Figure   2 Correlation between balls deformation and spin

Dimples

Dimples refer to the depressions on the surface of the ball, which have a significant effect on how far the ball travels.  Figure 3 shows the effect of dimples on a golf ball. 

Figure 3 The effect of dimples

 

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