The main problem discussed in this paper involves a simple one-dimensional two-body collision, in which the problem can be extended into a chain of onedimensional many-body collisions. The result is quite interesting, as it provides us with a thorough mathematical understanding that will help in designing a chain system for maximum energy transfer for a range of collision types. In this paper, we will show that there is a way to improve the kinetic energy transfer between two masses, and the idea can be applied recursively. However, this method only works for a certain range of collision types, which is indicated by a range of coefficients of restitution. Although the concept of momentum, elastic and inelastic collision, as well as Newton's laws, are taught in junior college physics, especially in Singapore schools, students in this level are not expected to be able to do this problem quantitatively, as it requires rigorous mathematics, including calculus. Nevertheless, this paper provides nice analytical steps that address some common misconceptions in students' way of thinking about one-dimensional collisions.

Moment of inertia is a very important property in the study of rotational mechanics. The concept of moment of inertia is analogous to mass in the linear motion, and its calculation is routinely done through integration. This paper provides an alternative way to compute moments of inertia of rigid bodies of regular shape using their symmetrical property. This approach will be very useful and preferred for teaching rotational mechanics at the undergraduate level, as it does not require the knowledge or the application of calculus. The seven examples provided in this paper will help readers to understand clearly how to use the method.

In high school and undergraduate physics, magnetic forces are often introduced as not being capable of doing work. In this paper, we explore the truth of this statement, analyzing the work done in various physical scenarios, including the bending of a wire in an external magnetic field, as well as the interaction between magnetic dipoles. In particular, the formation of surface charges and thorough analysis of the Hall force are elaborated. This paper addresses certain misconceptions regarding magnetic forces and work, and is expected to be of interest to students and educators alike.

Moments of inertia (MOIs) are usually derived via substantial integration and may intimidate undergraduates without prior backgrounds in calculus. This paper presents an intuitive geometric operation, termed “squashing,” that transforms an object into an equivalent one with a reduced dimension, whose MOI is simpler to determine. The combination of squashing and other methods (e.g., scaling arguments, the perpendicular-axis and parallel-axis theorems) enables the computation of complex MOIs with minimal integration.

Bertrand’s theorem is essential to the discussion of the motion of particles under central force. The theorem states that among all central forces, the only two in which bound orbits are also closed orbits are the harmonic force and inverse-square force. As existing proofs of the theorem require advanced mathematics, this paper presents a much more simplistic proof of Bertrand’s theorem without having to solve any complicated integrals or using mathematical tools beyond the scope of high school curriculum. The first part of the proof shows that the radial force must be a single power law, and the second part of the proof — which is the main crux of this paper — uses the fact that the radial distance of the motion must be a power of a sinusoidal function of the polar angle in nature. With a simple mathematical manipulation, the proof is then completed. This derivation is expected to be of great interest to high school teachers and students as it enhances the understanding of this distinct feature of inverse square force and harmonic force.