For the fourth problem of the 39th National Secondary School Physics Competition Repechage (Extended Examination), the force analysis of the four-ball system is first performed to obtain nine (non-linear differential) equations based on dynamics and kinematics to obtain two first integrals (conserved quantities). The solution is then given by numerical simulation. It is found that the original reference solution is a good approximation when the initial angular velocity is not large. Finally, the approximate solution is further investigated to obtain results accurate to t^5 in the Taylor expansion. The whole analysis provides a reference case for dealing with mechanics problems.

In high school textbooks, “the law of universal gravitation is derived from Kepler's law” and “Kepler's law is derived from the law of universal gravitation”, which are solved by approximate method, and the result is not convincing. In university textbooks or related literature, it is solved by calculus knowledge, which is not easy for students to understand. In this paper, Hamilton velocity vector diagram and simple geometric knowledge are used to skillfully solve the above problems.