拉普拉斯变换求解转动参考系下的真实自由落体

A Study of the Real Motion of Freely Falling Bodies with Laplace Transform in Rotating Frame of Reference

以地球北半球上空真实的自由落体运动为研究对象,基于牛顿第二定律建立了三维空间转动参考系下的二阶线性微分运动方程组,采用动量和拉普拉斯变换法求解得到真实自由落体的运动表达式;真实自由落体运动表达式在一定条件下可转换为理想的自由落体运动表达式,通过作图显示真实的自由落体运动轨迹是一条抛物线。此方法提供了一种简便、适应范围广的求解二阶线性微分方程组的方法,并有助于学生更直观地认识高空真实的自由落体运动。

This paper reports real motion of bodies freely falling in the earth＇ s northern hemisphere using Laplace transform method, which includes the influence of the earth＇s rotation and the air resistance. A two order linear differential equations of motions was established based on Newton＇s second law in the three-dimensional rotation reference flame, then solved by the Laplace transform method which we find is simpler for solving such equations, in which using momentum as intermediate variables can greatly simplify the calculation process, so that it only need to consider the initial velocity to solve the differential equations of motion, without removing two order infinitesimal amount and be independent on the origin of coordinates. The real freely falling motion expression can be converted to the ideal one under certain conditions. The plots show the pathway of the real freely falling body is a parabolic curve. This work pro- vides a simple and rational methods for solving two order linear differential equations, and gives the people a more intuitive understan- ding of the motion of bodies freely falling at a great height.