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

The diagrammatic approach to the collision problems in Newtonian mechanics is useful. We show in this article that the same technique can be applied to the case of the special relativity. The two circles play an important role in Newtonian mechanics, while in the special relativity, we need one circle and one ellipse. The circle shows the collision in the center-of-mass system. And the ellipse shows the collision in the laboratory system. These two figures give all information on two dimensional elastic collisions in the special relativity.


Introduction
Collisions of the interacting particles have fundamental importance in physics.
We often use the accelerated particles to investigate the substances.Cosmic ray which is often accelerated up to almost the speed of light collides with other particles in the air.For those particles which have high energy, special relativity has to be considered to investigate the collisions.

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].The two circles played an important role in Newtonian mechanics, while in the special relativity one circle and one ellipse play a crucial role.When the speed of the projectile tends to small compared to the speed of light, the ellipse becomes a circle and the Newtonian case recover in

Elastic Collision between Two Particles in Two Dimensions
Let us take a look the two dimensional elastic collision for later use. Figure 1 shows the collisions from the point of view in the laboratory and center-of-mass system and also show the notation which we use in this article.The projectile A has mass A m and the velocity A v and the target B has mass B m and the velocity B v before the collision.These quantities are known parameters or initial conditions in the laboratory system.The velocities after the collision are distinguished by the primes.And the asterisk is attached to the parameter in the center-of-mass system.In this article, we restrict ourselves that the target particle is at rest B = v 0 in the laboratory system before the collision.
The relation between the laboratory and center-of-mass systems is governed by the Lorentz transformation [6].Let V c β = be the relative velocity between two systems and is given by where c is the speed of light.The momentum A p is defined by its velocity A v as ( ) and the energy is given by ( ) The γ-factor is obtained by From the Lorentz transformation, the momentum of the incident particles in the center-of-mass system is given by , 2 where note that the momenta in the center-of-mass system are the same in magnitude after the collision: In the same way as the Newtonian mechanics [3], let ( ) cos ,sin θ θ * * * = n be the scattering angle of the projectile after the collision in the center-of-mass system.Since the angle θ * is not determined by the conservations of energy and momentum, we fix it according to the collision problems.Let ( ) x y -components of the momentum of the projectile in the laboratory system after the collision.The Lorentz transformation gives the relation between the laboratory system and center-of-mass system as follows: cos , where From these equations and the relation 1.
This equation indicates the ellipse [4] whose parameters minor semiaxis , 2 , 2 , 2 are uniquely determined by the initial conditions of the collision.The energy of the target in the center-of-mass system is defined by which is the same in magnitude before and after the collision.

Diagrammatic Technique
In this section, we deduce all relations, which we recalled in the former section, from the diagrammatic technique.Next, as shown in Figure 3, we draw an ellipse according to Equation ( 6) with the parameters from Equations (7) to (10).The point E is the midpoint of the foci E' and E".This ellipse signifies , and we find from Equations ( 8) and (10) that is the momentum of the projectile in the laboratory system before the collision.
Next, as depicted in Figure 4, we draw a broken line from the point C in parallel to the p x -axis until the broken line intersects with the ellipse.We call this point of intersection as F.Then, the vector A ′ = OF p becomes the momentum of the projectile A after the collision.The angle FOG θ ∠ = is the scattered angle of the particle A in the laboratory system.We note that the angel θ * in Figure 2 and the angle θ in Figure 4 are related each other.Once the θ * is fixed by the given collision problems, the θ is determined according to the prescription stated above.And the converse is also true.If the collision problem   6) can be drawn according as the parameters from Equations ( 7) to (10).The solid circle E' and E" on the p x -axis are the foci of this ellipse.
The point E is the midpoint of the foci.gives the angle θ in the laboratory system, we first draw the vector  in Figure 5, the projectile A can be deflected only through an angle not exceeding max θ from its original direction.This case is shown Figure 6.The maximum value of max θ is determined by the position F at which OF is a tangent to the ellipse.

Identical Particles and Newtonian Limit
The case      The semiaxes become the same length and the eccentricity tends to zero.The case of the Newtonian collision problems [3] [5] is recovered in this limit.

Conclusion
We derive the diagrammatic presentation of the two dimensional elastic collision problem in the special relativity.We draw the circle for the center-of-mass

Figure 1 .
Figure 1.Left: Collisions in the laboratory system.Right: Collisions in the center-of-mass system.

Firstly
, we draw a dashed circle whose radius is A , as depicted in Figure 2. The dashed circle shows the collision in the A. Ogura DOI: 10.4236/wjm.2018.89026356 World Journal of Mechanics

Figure 2 . 4 A=
Figure 2. Collision in the center-of-mass system.The figure shows that the masses 4 A m c = and 6 B m c = and the velocity 0.9 A v c = denote 3.892 p * = .The vectors

Figure 3 .
Figure 3.The ellipse Equation (6) can be drawn according as the parameters from

Figure 4 ..
Figure 4.The ellipse implies the collision in the laboratory system in case of the incident velocity 0.9 A v c = .The initial condition is as the same with Figure 2. The solid circles on the p x -axis denote the foci of the ellipse whose coordinates are 6.144 x p =

.
the ellipse.Then, we trace from F to C along the broken line.The vector A * ′ = OC p shows the momentum of the projectile A in the center-of-mass system.And the angle COA θ * ∠ = is the scattered angle of this system.Next, the vector B ′ = = FG OH p shows the momentum of the target B in the laboratory system after the collision.The angle FGO GOH φ collision.The ellipse has or has not intersections with p y -axis, according as A It is found from the magnitude of p γ * and A E c βγ * in Equations (8) and (10).The corresponding diagrams are shown in Figure4and A. Ogura DOI: 10.4236/wjm.2018.89026358 World Journal of Mechanics

Figure 5 .
Figure 5.As we see from Figure 4 that if A

4 A m c = and 3 B
Figure 5.If A

Figure 6 .
Figure 6.The initial condition is as the same with Figure 5.If A

Figure 7 .Equations
Figure 7.The incident particles have the same mass

Figure 8 .
Figure 8.The incident particles have the same mass

Figure 9 .
Figure 9.The incident particles have the same mass system and the ellipse for the laboratory system.Those circle and ellipse show the whole story of the two dimensional elastic collisions.When we use the graph paper for drawing those figures, we are able to measure the length of momentum vectors and the scattered angle by using the ruler and the protractor.This diagrammatic technique can help us understand collision problems qualitatively and quantitatively.