Team # 175 Page 3 of 8
1 Introduction
Although a hitter might expect a model of the batâ
˘
A¸Sbaseball collision to yield insight into
how the bat breaks, how the bat imparts spin on the ball, how best to swing the bat, and so on,
we model only the sweet spot.
There are at least two notions of where the sweet spot should beâ
˘
A
ˇ
Tan impact location on
the bat that either
• minimizes the discomfort to the hands, or
• maximizes the outgoing velocity of the ball.
We focus exclusively on the second definition.
• the initial velocity and rotation of the ball,
• the initial velocity and rotation of the bat,
• the relative position and orientation of the bat and ball, and
• the force over time that the hitteræŁ´r hands applies on the handle.
We assume that the ball is not rotating and that its velocity at impact is perpendicular to the
length of the bat. We assume that everything occurs in a single plane, and we will argue that
the hands?interaction is negligible. In the frame of reference of the center of mass of the bat,
the initial conditions are completely specified by
• the angular velocity of the bat,
• the velocity of the ball, and
• the position of impact along the bat.
The location of the sweet spot depends not on just the bat alone but also on the pitch and on
the swing. The simplest model for the physics involved has the sweet spot at the center of per-
cussion [Brody 1986], the impact location that minimizes discomfort to the hand. The model
assumes the ball to be a rigid body for which there are conjugate points: An impact at one will
exactly balance the angular recoil and linear recoil at the other. By gripping at one and impact-
ing at the other (the center of percussion), the hands experience minimal shock and the ball
exits with high velocity. The center of percussion depends heavily on the moment of inertia
and the location of the hands. We cannot accept this model because it both erroneously equates
the two definitions of sweet spot and furthermore assumes incorrectly that the bat is a rigid
body. Another model predicts the sweet spot to be between nodes of the two lowest natural
frequencies of the bat [Nathan 2000]. Given a free bat allowed to oscillate, its oscillations can
be decomposed into fundamental modes of various frequencies. Different geometries and ma-
terials have different natural frequencies of oscillation. The resulting wave shapes suggest how
to excite those modes (e.g., plucking a string at the node of a vibrational mode will not excite
that mode).
Theorem 1.1. L
A
T
E
X
Lemma 1.2. T
E
X.
Proof. The proof of theorem.
