Suppose that a moving ball is not spinning at all, and is orientated such that one of its stitches is exposed on one side, whereas the other side is smooth. It follows, from Fig. 11, that the drag force on the smooth side of the ball is greater than that on the stitch side. Hence, the ball experiences a lateral force in the direction of the exposed stitch. Suppose, now, that the ball is rotating slowly as it moves towards the hitter. As the ball moves forward its orientation changes, and the exposed stitch shifts from side to side, giving rise to a lateral force which also shifts from side to side. Of course, if the ball is rotating sufficiently rapidly then the oscillations in the lateral force average out. However, for a slowly rotating ball these oscillations can profoundly affect the ball's trajectory.

Watts and Sawyer^{23} have
performed wind tunnel measurements of
the lateral force acting on a baseball as a function of its angular orientation.
Note that the stitches on a baseball pass any given point *four* times for each
complete revolution of the ball about an axis passing through its centre.
Hence, we expect the lateral force to exhibit four maxima and four
minima as the ball is rotated once. This is exactly what Watts and Sawyer observed.
For the case of a 65mph knuckleball, Giordano has extracted the following
useful expression for the lateral force, as a function of angular orientation, ,
from Watts and Sawyer's data:

(57) |

Using the above expression, the equations of motion of a knuckleball
can be written as the following set of coupled first-order o.d.e.s:

(58) | |||

(59) | |||

(60) | |||

(61) | |||

(62) | |||

(63) | |||

(64) |

Note that we have added an equation of motion for the angular orientation, , of the ball, which is assumed to rotate at a fixed angular velocity, , about a vertical axis. Here, we are neglecting the Magnus force, which is expected to be negligibly small for a slowly spinning pitch. Appropriate boundary conditions at are:

(65) | |||

(66) | |||

(67) | |||

(68) | |||

(69) | |||

(70) | |||

(71) |

where is the initial speed of the pitch, is the pitch's initial angle of elevation, and is the ball's initial angular orientation.

The above set of equations have been solved numerically using a fixed step-length, fourth-order
Runge-Kutta method. The step-length, `h`, is conveniently expressed as a fraction of the
pitch's estimated
time-of-flight, , where
is the horizontal
distance between the pitcher and the hitter. Thus, `1/h` is approximately the number
of steps used to integrate the trajectory.

Figure 19 shows the lateral displacement of a set of four knuckleballs thrown with the same rate of spin, rpm, but starting with different angular orientations. Note the striking difference between the various trajectories shown in this figure. Clearly, even a fairly small change in the initial orientation of the ball translates into a large change in its subsequent trajectory through the air. For this reason, knuckleballs are extremely unpredictable--neither the pitcher, the hitter, nor the catcher can be really sure where one of these pitches is going to end up. Needless to say, baseball teams always put their best catcher behind the plate when a knuckleball pitcher is on the mound!

Figure 20 shows the lateral displacement of a knuckleball
thrown with a somewhat higher rate of spin than those shown previously: *i.e.*, rpm.
Note the curious way in which the ball ``dances'' through the air. Given that the
difference between a good hit and a bad hit can correspond to a shift in the strike point
on the bat by as little as of an inch, it must be quite a challenge to hit such
a pitch well!