Introduction

Rationale

A few months ago, I discovered a relatively new sport called Xingluo Ball, which is played on a sloped table with a uniquely designed target called the Xingpan. The goal of the game is to roll a large ball up a 28° inclined surface and accurately hit the target, which consists of four parallel pillars with only a 7 cm gap between them. As I experimented with launching the ball from different positions and speeds, I became increasingly curious about what makes a successful shot and whether it was possible to mathematically predict or optimize such a launch.

Given the constraints of the game — the slope of the table, the position of the target, and the ball’s size (5.22 cm in diameter) — I realized this problem offered an excellent opportunity to apply mathematical modeling, particularly in the context of projectile motion and coordinate transformations. Since I’ve already studied functions, trigonometry, and basic calculus in the IB Mathematics AA HL course, I wanted to apply these concepts to analyze the motion of the ball, and ultimately determine the most efficient way to score.

This exploration reflects not only my academic interest in mathematics but also my passion for sports design and motion analysis.

Aim

The aim of this exploration is to determine the optimal launch conditions — specifically, the launch angle and velocity — for a large ball rolling along a ° inclined table in order to successfully pass through a 7 cm gap between two vertical pillars located 3.65 meters away on the Xingpan. This will involve modeling the ball’s trajectory using parabolic and trigonometric functions, and identifying the viable set of parameters that lead to a successful shot.

Research plan

First, define all relevant measurements and geometric constraints of the table, the target, and the ball.

Second, construct a coordinate system and develop a mathematical model of the ball’s motion along the sloped surface.

Third, apply trigonometric functions and projectile motion equations to express the path of the ball as a function of launch angle and speed.

Then, solve for the conditions under which the ball intersects the narrow target region.

Lastly, evaluate the feasibility of the theoretical model by comparing the optimized parameters with the physical realities of the sport and suggest refinements if needed.