Evaluation of Creating Scoring Opportunities through Trajectory Analysis for Goal Scoring in Football

Determining the optimal launch angle for penalty kicks has practical value for coaches and performance analysts, yet the reviewed literature did not identify a computational study that applied projectile motion modelling to establish angle-specific thresholds under standard FIFA constraints simultaneously for both vertical and horizontal success criteria. This study analysed the trajectory of a football during penalty kicks using Newtonian mechanics and MATLAB simulation, testing eight launch angles (? = 10° to 45°) at a fixed initial velocity of 24.3 m/s. For each angle, maximum vertical displacement (ymax), ball height at the goal line (y??), and total horizontal range (x) were computed analytically using equations derived from Newton’s Second Law and verified numerically in MATLAB. The goal-scoring criterion required simultaneous satisfaction of two conditions: y?? < 2.44 m (below the FIFA-standard crossbar at x = 11 m) and x > 11 m (beyond the penalty mark). Only angles of 10° and 15° satisfied both criteria. At 10°, ball height at the goal line was 0.90 m, providing a 1.54 m safety margin below the crossbar; at 15°, goal-line height was 1.87 m with a 0.57 m margin. All six remaining angles (20° to 45°) violated the vertical criterion, with goal-line height rising from 2.87 m at 20° to 8.99 m at 45°. The horizontal criterion was satisfied at all eight angles, confirming that horizontal reach is never the binding constraint at this velocity. Within the scope of the reviewed literature, a contribution of this study is the explicit analytical separation of ymax (peak trajectory height) from y?? (ball height at the goal line), which produce different practical conclusions. These findings establish a quantified angular window (0° < ? ? 15°) for penalty kick success, provide coaches with an evidence-based training target, and demonstrate the utility of computational physics modelling as a tool for sports performance analysis.