golf ball flight

If you have played golf at least once in your life, you must have noticed that sometimes (or *sigh* often) the golf ball's pathline isn't straight but banana-shaped : it starts well but slowly shifts to the left or to the right -- and no, this is not because of the wind. Good golfers use these effects, known as draw when the ball is deflected to the left and fade when it is deflected to the right, to avoid obstacles.

Even if the player manages to send the ball straight in front of him, viewed from the side the pathline does not look at all like a parabola which is what that of an object in free fall should look like. In fact, Newton's second law of motion and the model of a pointlike object in free fall aren't enough to explain this behaviour. This is where fluid dynamics enter the room.

When halfway through the swing the golf ball is hit, it is transferred an amount of kinetic energy from the club which gives it its velocity. The instant velocity of an object is a vector (an arrow) that indicates in which direction it is moving. To determine the pathline of an object, three things must be known : its initial position, its initial velocity vector and all the forces applied. In fact the third unknown is its acceleration vector but Newton's second law of motion states that mass * acceleration = force.

An object in free fall

Say only the ball's weight is applied to it. This force is vertical and directed towards the ground and hence it is only responsible for the ball being attracted to the ground, that is to say it only modifies the vertical component of the velocity vector. In this model, the ball's pathline is included in a vertical plane and it is parabola shaped.

Here is a more scientific proof : By definition, the velocity is the time (t) derivative of position and the acceleration is the time (t) derivative of the velocity, this means that acceleration = d(velocity)/dt hence d(velocity) = dt * acceleration. The change in velocity d(velocity) is proportional to the acceleration. The velocity dt seconds later is old_velocity + d(velocity) = old_velocity + dt * acceleration. Since the acceleration vector is constant, by integration we get something of the form velocity = initial_velocity + acceleration * t and for the position something of the form position = initial_position + initial_velocity * t + 1/2 * acceleration * t ² which is the equation of a parabola.

acceleration :     velocity at t :   velocity at t+dt :
                           
     |                  _-7                      |  dt * acceleration
     |                _-                    _.7  V
     |              _-                   _.¨
     V             -                   .¨
     
The ball's pathline :

           __.....¨¨¨¨¨¨¨¨.....__
     _..--¨                      ¨--.._ 
  .-¨                                  ¨-. 
 ¨                                        ¨ 

This does not explain why fade and draw effects are possible and it predicts parabola shaped pathlines. Air resistance must also be taken in consideration. But since air resistance is a force opposed to the velocity it does not affect the direction of the velocity vector but its module : the ball is slowed down.

The force applied to an object parallel to the relative wind is called the drag force. It can be calculated with the formula : drag_force = 1/2*C_D*ρ*U²*L² where ρ is the air density, U is the velocity, L is the scale of the body and C_D is called drag coefficient and is around 0.3 for golf balls. Since the acceleration, which is the velocity derivative depends on the velocity, finding the pathline of the ball is harder because it involves solving a differential equation. Nevertheless here is more or less what it looks like :

The ball's pathline with drag force :

           __.....¨¨¨¨¨¨¨..._
     _..--¨                  ¨-_ 
  .-¨                           . 
 ¨                               - 

This pathline happens to be fortunate for the golfer, despite its shortening the range : just before landing, the direction is nearly vertical which prevents the ball from bouncing and rolling in a random direction. The ball comes to a halt in a zone very close to the spot it first landed on.

Fade and draw

In fact, when the golf ball does a fade or a draw, it rotates on itself very fast. The speed of air molecules surrounding the ball is affected, creating areas of greater and lower pressure which attract the ball. This effect is known as the Bernoulli Effect and also explains why airplanes can fly.

When the ball rotates on itself, its relative speed on one side is greater than that on the other because velocities add themselves. The rotation entrains air molecules and the difference in speed is responsible for pressure variations. Bernoulli's equation (simplified) states that P + 1/2*ρ*v² is constant (called dynamic pressure), ρ is the air density and v the air speed. Hence where the air speed is greater, the pressure is lower.

       ball rotating clockwise
       and moving to the left
        viewed from the top

                                    ^ lift
       air speed is greater         |
w -->    lower pressure       =>  _.|._
i -->                            ...|...
n -->                            ....--------> drag
d -->  air speed is lower     =>  ¨...¨
        greater pressure

The pressure difference creates a suction force perpendicular to flow of air called the lift force. It can be calculated with the formula : lift_force = 1/2*C_L*ρ*v²*α*A, where C_L is the lift coefficient, A is the area of the ball, α is the angle of attack and v is the ball's velocity.

If the ball is launched with a positive rotation speed around the vertical axis (viewed from the top it rotates counterclockwise), the air speed at the left of the ball is higher than that at the right, therefore the pressure is lower, hence the ball is attracted to the left. This is called a draw. When the ball is attracted to the right it is called a fade.

Backspin

In fact when the club hits the ball it makes it rotate very fast around the horizontal axis (50 rev/s) This is called backspin because the top of the ball spins towards the back of the pathline. And as a consequence, the speed of air on the top is higher than that on the bottom : the ball is attracted towards the sky. It overrides gravity's effects for a moment resulting in an almost straight course.

Air resistance slows the ball's rotation speed and backspin is only an important factor up to the 2/3 of its course. This explains why, as shown on the below diagram, the first part of the ball's pathline is an almost straight line.

The ball's pathline with drag force :

                ___---¨¨¨¨¨¨¨..._
          ___---                 ¨-_ 
    ___---                          . 
 ---                                 - 

The dimples on the ball are very important too. Why ? The short answer is that a dimpled spinning ball has a more intense turbulent boundary layer around it than a soft one, which reduces the drag force.

Closing words

These are the most important factors that govern a golf ball's course, except that I've left over the importance of dimples. If you wonder Why are Golf Balls Dimpled?, then read it ! UPDATE: this node seems to have disappeared, 2004/09/24

Of course this is also true for other sports : ping-pong, tennis, baseball...