Falls remain a leading health risk among the elderly, demanding accurate and unobtrusive detection technologies that perform reliably in real-world conditions. This study introduces a vision-based fall detection framework that integrates real-time pose estimation with polynomial temporal modeling of body ratio dynamics. The system employs YOLOv8n-pose to track skeletal keypoints from video streams, then derives body height-to-width ratios across frames. These temporal sequences are transformed using sixth-degree polynomial fitting, yielding compact feature representations of motion patterns. A fine-tuned k-Nearest Neighbors (KNN, k = 10) classifier has been used and reached an accuracy of 97.4%, alongside 100% specificity and 94.74% sensitivity. The effectiveness of the framework was confirmed using the ROSE Lab dataset, revealing significant speed, strength, and flexibility across a variety of settings. This approach offers a scalable, non-intrusive, and privacy-conscious solution for real-time fall detection in healthcare and assisted living environments.