Optimal strategies for throwing accurately

TL;DR

This paper develops a mathematical framework to analyze and predict optimal throwing strategies by quantifying how errors propagate through projectile motion, considering different throwing styles and target configurations.

Contribution

It introduces a Jacobian-based method to predict preferred throwing styles and characterizes the trade-off between speed and accuracy in projectile throwing.

Findings

Underarm throws tend to undershoot targets.

Overarm throws do not necessarily undershoot.

Optimal shooting angles bifurcate based on initial condition noise.

Abstract

Accuracy of throwing in games and sports is governed by how errors at projectile release are propagated by flight dynamics. To address the question of what governs the choice of throwing strategy, we use a simple model of throwing with an arm modelled as a hinged bar of fixed length that can release a projectile at any angle and angular velocity. We show that the amplification of deviations in launch parameters from a one parameter family of solution curves is quantified by the largest singular value of an appropriate Jacobian. This allows us to predict a preferred throwing style in terms of this singular value, which itself depends on target location and the target shape. Our analysis also allows us to characterize the trade-off between speed and accuracy despite not including any effects of signal-dependent noise. Using nonlinear calculations for propagating finite input-noise, we find that an underarm throw to a target leads to an undershoot, but an overarm throw does not. Finally, we consider the limit of the arm-length vanishing, i.e. shooting a projectile, and find that the most accurate shooting angle bifurcates as the ratio of the relative noisiness of the initial conditions crosses a threshold.