Based on experimental materials for a simple pendulum that provide only chart images and include no raw numerical data files, this paper distinguishes theoretical derivations from qualitative image-based observations and explores how time series can be used to represent calculus relationships in periodic motion. The experimental setup described in the materials uses the Arduino Nano 33 IoT development board, the LSM6DS3 inertial measurement unit, and Bluetooth transmission, covering five pendulum lengths from 0.34 m to 1.45 m, with an approximate sampling frequency of 100 Hz. Based on the small-angle simple pendulum model, the analysis establishes the relationship among angular displacement, angular velocity, angular acceleration, period and frequency, and examines the oscillation interval, release shock, decay behavior and multi-axis response in the records. The results show that the images qualitatively support the theoretical trend of "longer pendulum length, slower oscillation", but they are not enough to verify the measured frequency, damping constant or three-dimensional displacement. The study also illustrates how differentiation amplifies noise and why double integration depends on calibration, initial conditions, and drift control. This case provides an intuitive example for establishing links between functions, derivatives, integrals, differential equations, and observable motion characteristics in physics teaching.

Through geometric optimization, airfoil designs directly affect efficiency, stability, and energy performance in aerospace and wind systems. More specifically, airfoil geometry optimization improves aerodynamic efficiency and reduces energy losses in aircraft and wind turbines, thus supporting technological innovation and cleaner energy utilization. This paper examines the physical mechanisms of lift generation and evaluates the influences of key airfoil geometric parameters, including maximum thickness, maximum camber, their chordwise locations, and trailing-edge configurations, on lift performance. To explain these effects, the analysis examines lift mechanisms from classical inviscid circulation models, including the Kutta-Joukowski theorem and thin-airfoil theory, to viscous-flow frameworks describing boundary-layer transition, shear stress distribution, and vorticity dynamics. At low Reynolds number conditions (0.3×10⁵ to 0.8×10⁵), viscous effects become significant and produce laminar separation bubbles that reduce aerodynamic efficiency. The results show that airfoil geometric parameters regulate lift generation mainly by reshaping surface pressure distributions and modifying boundary-layer stability, providing a physics-based foundation for optimized lifting surface design.