The motion of particles on a smooth hemisphere is a good research problem in high school. In this article, we extend this problem to any rough surface and derive the explicit integral form of velocity based on Newton's second law. It has an analytical expression in some cases and obtains a universal expression to determine the separation angle condition for particles leaving the surface, In the end, three curves, a circle, a parabola, and an ellipse, were analyzed in detail using Mathematica software, and several interesting conclusions were obtained.

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.