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

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.


Introduction
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. Then, we want to find the final states, especially, momenta and energies after the collision. 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]. Be-How to cite this paper: Ogura, A. (2020) Collisions in Classical Mechanics in Terms of Mass-Momentum "Vectors" with Galilean sides, it is also important to obtain the relation between the laboratory and the center-of-mass systems. It is complicated to understand the collision problems in classical mechanics.
The purpose of this paper is that we resolve the complexity of the collision problems and we show a new looking at them. In recent years, the mass-momentum diagram is used to analyze the collision problems in Newtonian mechanics [2] [3]. There, the mass-momentum "vectors" are introduced. These diagrams show the whole story of the collision problems in both the laboratory and the center-of-mass systems. However, the discussion of the paper [3] still stuck to the simultaneous equations. 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. This paper also gives the theoretical background of them. Now consider two reference frames K and K'. We assume that the frame K' moves in the x-direction at speed V with respect to K. And let us assume the origins O and O' of the two reference frames coincide with time 0 t = . An event that occurs at some point is observed from both frames, which are characterized by a set of coordinates ( ) , , , t x y z and ( ) , , , t x y z ′ ′ ′ ′ . The Galilean transformations give the relation between two coordinates and it is described by The inverse transformation is given by just putting −V to V in Equation (1). In the following paper, we designate the frame K as the laboratory system, while K' as the center-of-mass system. Accordingly, the velocity V describes the velocity of the center-of-mass.
In this article, we show that the collision problems are solved by using mass-momentum "vectors" ( ) , , , x y z m p p p and this Galilean transformation.
Our strategy and notations are pictorially stated in Figure 1. 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. See the dashed arrow in Figure 1. However, we propose that we make a detour through the center-of-mass system for obtaining the final states. It is applicable to both one and two dimensional collisions. Our strategy is as follows.  Figure 1. We also attach the prime for the variables after the collision. In this frame, two particles move in the opposite direction after the collision with the same magnitude of momentum. We introduce the coefficient of restitution e for one dimensional collisions and the collision angle θ * of the incident particle for two dimensional elastic collisions.  Figure 1. Finally, we reach the final states. We never solve the simultaneous equations in contrast with the usual treatment of the collision problems.
5) Let us consider the two special cases. One is that the target particle is at rest ( 0 B p = ) in the laboratory system before the collision.
6) The other is that, in addition to the case ( 0 B p = ) above, two particles have This paper is organized in the following way. In Section 2, we discuss one dimensional collision. The role of the coefficient of restitution e is clearly stated.
We also discuss the case in which the target particle is at rest before the collision and two particles have equal masses. In Section 3, we turn to the two dimensional elastic collision. 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.

Collisions in One Dimension
Let us discuss the one dimensional collisions. The motions of the particles are restricted in the x-direction. Therefore, the yand z-components of the mo-

Velocity of Center-of-Mass System
We discuss the strategy 1 in the Introduction. Consider the Galilean inverse transformation with the whole two body system, where, 0 A B p p * * + = is the definition of the center-of-mass system. From the second row of the matrix, we obtain the velocity of the center-of-mass, which is conserved throughout the collision because of the conservation law of mass and momentum in Newtonian mechanics.

Momentum in the Center-of-Mass System before the Collision
We discuss the strategy 2 in the Introduction. Concerning the Galilean transformation for each particle, we obtain the momenta in the center-of-mass system before the collision; , where, we used Equation (3). It is natural that 0 of the definition of the center-of-mass system. Then we define a momentum p * as for later use. Note that the momentum p * is written in terms of velocity as Energies in the laboratory system before the collision Before concluding this subsection, we write down the energies in the laboratory system before the collision in terms of p * . From Equations (5) and (6), we obtain , .
Thus, the kinetic energies in the laboratory system before the collision are given by and their sum becomes where, we used Equation (3). The first term of the right hand side shows the kinetic energy of the center-of-mass, while the second term shows the kinetic energy of the relative motion with respect to the center-of-mass.

Momentum in the Center-of-Mass System after the Collision
We discuss the strategy 3 in the Introduction. We determine the momenta in the center-of-mass system after the collision. In this frame, the particles move in the opposite direction. We introduce the coefficient of restitution e ( 0 1 e ≤ ≤ ), and write down the momenta in the center-of-mass system after the collision , .
Note that these momenta can be derived from the usual definition of the coefficient of restitution: where, A E * and B E * are given by Equations (8) and (9). Namely, the kinetic energies in the center-of-mass system are decreased by 2 e .

Momentum in the Laboratory System after the Collision
We discuss the strategy 4 in the Introduction. Consider the Galilean inverse transformation for each particle, we obtain the momenta in the laboratory system after the collision. From the second row of these matrices, we obtain ( )

,
The second terms of the right hand side are interpreted by the momentum lost by particle A and the momentum gained by particle B, which are called impulse in Newtonian mechanics. Adding the two equations, we easily see the con- the first term, which is the kinetic energy of the center-of-mass, does not change, while the second term, which is the kinetic energy of the relative motion with respect to the center-of-mass, decreases by 2 e . By using Equation (13), we ob- The third term of the right hand side is called Q-value, which is larger than zero. Namely, the total energy of the whole system is decreased in Q-value.

In Case of B p 0 =
We discuss the strategy 5 in the Introduction. In this case, we clearly see the re- =.
In addition, we obtain the kinetic energies for each particle and energy conservation law and obtain the momenta in the laboratory system after the collision

In Case of
We also obtain the kinetic energies ( ) ( ) and their sum Here, 1 e = shows the elastic collisions. After the collision, the incident particle A stops and the initially rest particle B moves with the momentum of which the particle A had before the collision.

Elastic Collisions in Two Dimensions
Let us turn our discussion to the case of the two dimensional elastic collisions, i.e., the case in which the coefficient of restitution e is equal to 1.
We suppose that the motions of the particles are restricted in the x-y plain so that the z-component of the momentum is zero. Since the motions of the par- The diagrammatic technique of this section is fully discussed in [6].

Momentum in the Center-of-Mass System after the Collision
We discuss the strategy 3 in the Introduction. In the center-of-mass system, the magnitudes of the momenta do not change before and after the collision. Thus, we write down the momenta in the same way with Equation (7), However, the direction of the momenta changes in the two dimensional collision. As shown in Figure 2, we define the sense of the momentum A * Since the magnitudes of the momenta do not change in this frame after the collision, the kinetic energies of each particle do not change: where, A E * and B E * are the same with Equations (8) and (9).

Momentum in the Laboratory System after the Collision
We discuss the strategy 4 in the Introduction. The motion of the particles after the collision is supposed to occur in the x-y plain. Thus, the momenta are writ- , where θ and φ are the scattering angle of the particles A and B in the laboratory system as shown in Figure 2.
From the Galilean inverse transformation, we obtain the momenta in the laboratory system after the collision by using Equation

In Case of B p 0 =
We discuss the strategy 5 in the Introduction. The same condition of Equation