Abstract

We determine the sweet spot on a baseball bat. We capture the essential physics of the ball–bat impact by taking the ball to be a lossy spring and the bat to be an Euler-Bernoulli beam. To impart some intuition about the model, we begin by presenting a rigid-body model. Next, we use our full model to reconcile various correct and incorrect claims about the sweet spot found in the literature. Finally, we discuss the sweet spot and the performances of corked and aluminum bats, with a particular emphasis on hoop modes.

Model Overview

A cycle can be divided into two parts: movement on the half-pipe, and the airborne performance.

For the first, we focus on the conversion and conservation of energy. The loss of mechanical energy Elost due to the resistance of snow and air is the key. We derive a differential equation for it. We cannot neglect the snowboarder’s increasing the mechanical energy by stretching the body (standing up) and doing work against the centrifugal force.

To derive an expression for vertical air, we apply Newton’s Second Law. If we neglect air drag during the flight (we later show that it is indeed negligible), we can calculate vertical air, duration of the flight, flight distance, gravitational potential decrease, etc.

Next, we discuss the airborne rotation of the snowboarder. Since the shape of the half-pipe directly influences the initial angular momentum of the snowboarder, and the angular momentum cannot change during the flight, the relationship between the half-pipe shape and the initial angular momentum is the key to our discussion. After deriving an expression for the initial angular momentum, we can find the optimal shape of the half-pipe.

Introduction

The velocity of the ball leaving the bat is determined by

l  the initial velocity and rotation of the ball,

l  the initial velocity and rotation of the bat,

l  the relative position and orientation of the bat and ball,

l  the force over time that the hitter’s 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,

l  the velocity of the ball, and

l  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.

Following the center-of-percussion line of reasoning, how do we model the recoil of the bat? Following the vibrational-nodes line of reasoning, how do we model the vibrations of the bat? In the general theory of impact mechanics [Goldsmith 1960], these two effects are the main ones (assuming that the bat does not break or deform permanently). Brody [1986] ignores vibrations, Cross [1999] ignores bat rotation but studies the propagation of the impulse coupled with the ball, and Nathan [2000] emphasizes vibrational modes. Our approach reconciles the tension among these approaches while emphasizing the crucial role played by the time-scale of the collision.

Overall Approach

We employ a multi-step approach to finding an optimal solution to the problem. First, we tackle the requirement of sprinkling the entire field. Using geometrical analysis, we reduced this problem to a covering problem [3], which translates to finding the least number of equally-sized circles that can cover any given area. However, this solution results in the placement of sprinklers outside the field boundaries, and so we perturbed the solution to readjust the placement of the sprinklers, all the while maintaining the condition of complete coverage of the field. We then use this new solution as a blue print for finding the minimal number of pipe setups by experimentally "fitting" the pipe sets through the sprinklers (if possible). This is accomplished by using an algorithm that perturbs (in each iteration) the sprinkler layout, finds the minimum amount of pipe setups, perturbs the layout again (in the next iteration), finds the minimum amount of setups, and so on. After a specified amount of iterations, it then outputs the minimum of all the found minimum setups. The rationale for perturbation is that we are willing to sacrifice some uniformity in order to find the least number of setups, while simultaneously ensuring that we are still sprinkling the entire field. We then feed this layout of pipe setups to another algorithm which generates an irrigation schedule.