关于速度合成定理两种推导法的统一性分析

On the Unity of Two Deductive Methods of Velocity Synthesis Theorem

理论力学中,关于点的复合运动速度合成定理的几何法推导长期存在争议,严重影响该部分内容的教学。文章基于点的运动描述的矢量法,剖析了推证速度合成定理的解析法与几何法,根据运动分解的数学内涵,论证了几何法的合理性与严谨性,同时阐明了两种推导法在物理概念和数学推导方面的统一性。为了便于教与学,将两种方法分别归纳为运动耦合求导法和运动解耦分析法。

In theoretical mechanics, the derivation of velocity synthesis theorem of a particle by the geometric method has been controversial for a long time, which seriously affects the associated teaching.In this paper, two deductive methods, i.e., the analytical method and geometric method, for velocity synthesis theorem are analyzed based on the vector method of kinematics of a particle. In the light of mathematical connotation of motion decomposition, the reasonability and strictness of the geometric method are illustrated. Meanwhile, the unity of the two deductive methods in physical conceptions and mathematical deductions is demonstrated. In order to facilitate teaching and learning, the two methods are respectively referred to as differentiation of coupled motion and analysis of decoupled motion.