Abstract
We present the usefulness of mass-momentum “vectors” to analyze the collision problems in classical mechanics for both one and two dimensions with Galilean transformations. The Galilean transformations connect the mass-momentum “vectors” in the center-of-mass and the laboratory systems. We show that just moving the two systems to and fro, we obtain the final states in the laboratory systems. This gives a simple way of obtaining them, in contrast with the usual way in which we have to solve the simultaneous equations. For one dimensional collision, the coefficient of restitution is introduced in the center-of-mass system. This clearly shows the meaning of the coefficient of restitution. For two dimensional collisions, we only discuss the elastic collision case. We also discuss the case of which the target particle is at rest before the collision. In addition to this, we discuss the case of which the two particles have the same masses.
Highlights
Collisions of the interacting particles have fundamental importance in physics
We present the usefulness of mass-momentum “vectors” to analyze the collision problems in classical mechanics for both one and two dimensions with Galilean transformations
We show that just moving the two systems to and fro, we obtain the final states in the laboratory systems
Summary
Collisions of the interacting particles have fundamental importance in physics. To concerning collision problems in classical mechanics, it is customary that the initial states, for example the mass and momenta, are given. To obtain the final states, we have to solve the simultaneous equations of the momentum-conservation law and the definition of the coefficient of restitution [1]. This paper shows that we never solve them for obtaining the final states This diagrammatic approach is used for two-dimensional collision problems [4] [5] [6]. In the introductory textbooks of physics [1], we have to calculate the simultaneous equations of momentum-conservation with energy-conservation or the definition of the coefficient of restitution in order to obtain the final states. Attach the asterisk for the variables in the center-of-mass system In this frame, two particles make a head on collision with the momentum which have the same magnitude p∗.
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