投擲運動數學模型之研究

A Study for a Mathematical Model of the Projectile Motion

10.6222/pej.0021.199606.2112

林建德(Kin-Tak Lam)；姬梁文(Liang-Wen Chi)

投擲運動 ； 科學運動訓練 ； 最佳拋射角 ； 最大拋射距離 ； Projectile motion ； Scientific physical education ； Best angle of delivery ； Maximum distance of delivery

體育學報

21期（1996 / 06 / 01）

127 - 135

繁體中文

本文之主要目的是應用運動學基本原理，建立無阻抛體運動軌跡之數學模型。文中首先根據拋物體運動學的觀念，推導在忽略空氣阻力情況下拋射體抛射距離與抛射高度之運動方程式；然後應用隱函數微分理論，由所得之運動方程式求得最佳拋射角、最大拋射距離、拋射初速度與投擲出手之拋射高度的簡單關係式，並證實了當拋射高度與抛射初速度一定時，僅有一組最大拋射距離與最佳拋射角的解；最後利用微電腦由所得之關係式列出最大拋射距離與最佳拋射角的數值，以做為投擲運動（鉛球、鏈球）科學運動訓練的工具。

This paper presents a mathematical model of the projectile motion for physical education. Based on the principle of kinematics, the equation of projectile motion is developed. Then, by using differential calculus, a simple expression of relationships among the best angle of delivery, the maximum distances of delivery, the initial velocity of delivery, and the height of delivery point is introduced. According to this expression, we conclude that the best angle of delivery and the maximum distances of delivery are only related to the height of delivery and initial velocity of delivery. Finally, a diagram for the values of the maximum distance of delivery and the best angle of delivery are developed by computer. The results are used as a tool for the scientific physical education in the projectile motion (as shot put and hammer throw).