Diagrammatic Approach for Investigating Two Dimensional Elastic Collisions in Momentum Space I: Newtonian Mechanics

Abstract

We present the usefulness of the diagrammatic approach for analyzing two dimensional elastic collision in momentum space. In the mechanics course, we have two major purposes of studying the collision problems. One is that we have to obtain velocities of the two particles after the collision from initial velocities by using conservation laws of momentum and energy. The other is that we have to study two ways of looking collisions, i.e. laboratory system and center-of-mass system. For those two major purposes, we propose the diagrammatic technique. We draw two circles. One is for the center-of-mass system and the other is for the laboratory system. Drawing these two circles accomplish two major purposes. This diagrammatic technique can help us understand the collision problems quantitatively and qualitatively.

Collisions in Classical Mechanics in Terms of Mass-Momentum “Vectors” with Galilean Transformations

“…We introduce the collision angle θ * of the outgoing particle after the collision. We show the theoretical background for the diagrammatic approach [4] [5] [6]. Section 4 is devoted to a summary.…”

“…This paper shows that we never solve them for obtaining the final states. This diagrammatic approach is also used for two-dimensional collision problems [4] [5] [6]. There, we see the collision problems on the two-dimensional momentum space.…”

Elastic Collisions in Minkowski Momentum Space with Lorentz Transformations

“…For two dimensional collision in classical mechanics, two-dimensional momentum space describes the collision clearly in the textbook [3]. We also see the slightly different illustration which lays emphasis on the transformation of the two systems [4]. For one dimensional collision in the special relativity, Saletan [5] proposed to understand the collision problems in the Minkowski momentum space, with energy E/c represented along the vertical axis and momentum p represented along the horizontal axis.…”

Diagrammatic Approach for Investigating Two Dimensional Elastic Collisions in Momentum Space II: Special Relativity

“…Diagrammatic technique gives the powerful tool to investigate the collision in Newtonian mechanics [1] [2] [3]. In this article, we apply it to the relativistic collision problems [4] [5].…”

“…Here, we note that as we see from Equations (14)-(17), the different speed of the incident particle gives the different ellipse in magnitude. As the speed of the incident particle decreases, the eccentricity of the ellipse decreases and the ellipse begins to resemble a circle[3] [5]. It is found that if we take the limit with c → ∞ , the parameters in…”