Simulation modeling of a thrown ball bouncing nonlinearly across a grid of cups

摘要

This work is motivated by a staple carnival game. A player throws a ping-pong ball onto a grid of cups with the goal of having the ball land in a cup. Though there are many variations to this game, there is a common underlying characteristic. As the ball bounces on the cup grid, its sequence of bouncing trajectories becomes nonlinear. It is this nonlinearity which makes it impossible for an observer to predict the outcome, and makes the game difficult.The nonlinearity comes from the interaction of the ball’s linear motion, angular motion, and the how it bounces off the cup edges. The insight that led to the development of this model is that the ball bouncing on a cup edge is equivalent to it bouncing on a tilted surface. Thus, to develop a predictive model for this game, we modeled a spinning partially elastic ball as it bounces over a series of arbitrarily-tilted surfaces.We embedded this algorithm in a Monte Carlo simulation model which simulates a player throwing the ball while varying initial launch parameters. Using this model, we were able to track possible trajectories and make probabilistic statements about various outcomes of the game. Furthermore, we used our empirical results to suggest different scenarios for the game, then applied the model to assess and quantify their impacts on difficulty.Visual inspection and brief analysis of an actual game support our model’s credibility.