Bowling ball motion differential model

Abstract
Bowling game is not only an interesting and involving kind of sports but also it is a perfect physical and mathematical site. With the bowling game, we have an extremely seemingly simple situation: after the ball is released onto the lane its behavior is determined by a relatively small number of laws of mechanics. Their combined action results in interesting and entirely non-simple objective laws in the ball‘s behavior on its way to the target – the 10 pins specially arranged and the lane‘s end.

While the ball is moving along the lane it participates simultaneously in two kinds of motion: forward sliding and rotation interacting with each other to result in the ball‘s possible curvilinear trajectories. The ball‘s asymmetrical inner structure and varying from game to game the rules to apply oil onto the lane (the oiling pattern) make the game still more complicated and allow an experienced bowler to redistribute this curvature along the trajectory so as to provide the ball‘s most effective entrance in the pins area.

In this work, with reference to the Coulomb-Amontons law defining the sliding friction force, we setup a symmetrical ball motion mathematical model. This model is reduced to the second-order differential equation that allows for an interesting integral (sliding). Integrating this equation leads to the ball‘s parabolic trajectory whose position of vertex, direction of parabola axis and flatness of parabola branches are a complicated function of the initial conditions (ball release).

Based on the motion law general form we derive an explicit formula for the game parabola characteristics and their related non-dimensional parameters characterizing one or another game style, obtain explicit expressions for certain release parameters, and when the latter are not adhered to it results in ineffective bowling game – the track flare (formation of oil rings on the ball‘s surface).

This work is specific for a relative simplicity of the model and complexity of the effects that allow, with the use of the model, for a qualitative explanation and quantitative description. All the ball’s behaviour basic specifics discussed at bowlers training sessions with reference to the weak and inaccurate similarities and explanations, - we translate into a qualitative language of equations and formulas.

The work has a practical value since its results are the basis for a rational classification of the bowling game styles and allow bowling game purposeful planniing and correcting of individual tactics and techniques aimed to enhance the game effeciency.

Maxim Okorokov