Guan Keying

Department of Mathematics, School of Science, Beijing Jiaotong University

Email: keying.guan@gmail.com

Abstract: Under the framework of special relativity, this paper considers a basic problem of two-dimensional elastic collision, that is, if a high-speed moving particle 1 collides with another originally stationary particle 2, when these four parameters, namely the static mass of the two particles m₁ and m₂ , the impact velocity v and the scattering angle θ are all given, then what are the solutions of the unknown velocities v₁ and v₂ of the two particles after collision. With the help of the well-known mathematical software system <Wolfram Mathematica>, combined with some manual derivation skills, according to the laws of conservation of momentum and energy, the exact solutions of the unknown velocities v₁ and v₂  are obtained.  On the basis of these exact solutions, three important coefficients are established, namely, the scattering coefficient R_S of the impacting particle 1, the absorption momentum coefficient R_AM and the absorption kinetic energy coefficient R_AK of the impacted particle 2, which can well reflect the mechanical effect of the collision.  This paper also considers the extremely case that the static mass of the impacting particle 1 is zero and get the exact corresponding expressions. Despite the complexity of these exact expressions, some important conclusions can still be drawn from precise calculations, qualitative studies, and graphic analysis. Similarities and differences between these different elastic collision theories, including elastic collision in classical mechanics, are also discussed.

I.  Research background

Physics and mechanics are the most basic sciences for people to understand the material world. Wikipedia describes them as follows

Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. Physics is one of the most fundamental scientific disciplines, and its main goal is to understand how the universe behaves.

Mechanics (Greek: μηχανική) is the area of mathematics and physics concerned with the motions of physical objects, more specifically the relationships among force, matter, and motion.

An elastic collision is an encounter between two bodies in which the total kinetic energy of the two bodies remains the same. In an ideal, perfectly elastic collision, there is no net conversion of kinetic energy into other forms such as heat, noise, or potential energy. There is no doubt that elastic collision is the most basic research object in physics and mechanics.

Since the elastic collision of two particles is the simplest interaction, the causality of this mechanical action can be accurately described mathematically. Although the study of this interaction is relatively simple, the results of the study can provide an accurate and solid foundation for the study of more complex phenomena, such as the study of air pressure, light pressure, etc.

It is from the above considerations that, in order to understand light pressure, the author has studied the exact solution of classical elastic collision in the case of an impacting particle colliding with a primitive static particle (impacted particle) at an impact velocity u, and got the expression of the scattering coefficient RS of the impacting particle, and got also the expressions of  the absorption momentum coefficient R_AM and the absorption kinetic energy coefficient R_AK for the impacted particle. The subscripts _S, _AS and _KS of R_S, R_AM and R_AK refer the scattering, the absorbed momentum and absorbed kinetic energies respectively. The strict definition of these coefficients can refer to expressions (22), (23) and (24) in this paper. A detailed description of these quantities in classical mechanics can be found in my blog.

两质点完全弹性碰撞后的反射系数及相关的动量与能量分配--再论太阳帆不应使用全反光材料

Note: Various numbers of symbols and terminology used in this article have been improved and differ somewhat from the blog post above

These quantities can be used to understand (or study) the mechanism of the concept in light pressure or in air pressure, such as "reflection coefficient", the absorbed momentum and kinetic energy of the pressed matter.

Recently, the author noticed that, in the classic work "MECHANICS"^[1] of the famous former Soviet physicists  L. D. Landau and E. M. Lifshitz, this issues have been discussed partially, and essentially, it gives the same velocity distribution formula. But the velocity distribution is expressed by a particular reflection direction angle χ which is measured in the frame of reference of center of mass system, which is not convenient in actual use. In addition, article [1] did not discuss the problem about the "the momentum coefficient" and "captured kinetic energy coefficient" of the impacted particle.

While exploring the high-speed elastic collision of two particles in relativity, the authors also noticed that although the elastic collision of two particles is the simplest case, it is still not an easy mathematical task to obtain an accurate solution. So far, despite its obvious importance, the authors have not seen in the literature an exact solution for the relativistic elastic collision of two particles in two dimensions.

Therefore, the authors consider elastic collisions in relativistic mechanics and obtain some precise results, which will be presented in the following sections.

II.  The exact solution of relativistic elastic collision in 2D

Consider the relativistic two-dimensional elastic collision of two particles 1 and 2, their static masses are m₁, m₂, respectively. Suppose the velocities before the collision are u₁, u₂, and the velocities after the collision are v₁, v₂.  To simplify the problem, u₂ is assumed to be zero,  so the particle 2 is called the impacted particle.  Naturally, the velocity u₁ is called the impact velocity, abbreviated as u, and the particle 1 is called the impacting particle. Based on the conservation of momentum, it is obvious that the three velocity vector u, v₁ and v₂  should be distributed on a common plane. 

According to the relativity, the corresponding mechanical quantities of these two particles before collision, such as the energies, momentums, kinetic energies are denoted respectively as

    (1.1)

and            

     (1.2)

and the corresponding mechanical quantities of these two particles after collision are respectively as         

 (2.1)

and 

                         (2.2)

where  c is the speed of light, the subscripts _b and _a used in the above symbols refer to before and after the collision.  According to the theory of relativity, the following symbols are used

          (3)

the notations |u|, |v₁| and |v₂| represent the magnitudes of the corresponding vectors.

Clearly, the three non-zero momentun vectors P_1b, P_1a and P_2a are also distributed in the common plane (see Fig 1)

Fig 1

The angle θ between P_1b and P_1a is called the scattering angle of the impacting particle 1, and its value range is 

  (4)

and the angle φ between P_1b and P_2a is called the recoil angle of the impacted particle 2, and it can only take values within the following range

   (5)

In this elastic collision problem, the conservation of energy means that

  (6)

and the conservation of momentum means 

   (7)

Clearly,  when m₁, m₂, β and θ are given, the exact solutions of β₁ and β₂ can be obtained from (6) and (7). However, it is really not an easy thing to do by manual derivation. 

Fortunately, with the help of Wolfram Mathematica, a well-known modern mathematical software system, the author accomplished this task with some manual derivation skills.

Let 

        (8)

then β₂² can be represented by r, β and β₁ from  (6)

(9)

Transforming the vector equation (7) into its component form, and using (4), we get

    (10)

and

     (11)

From (10) and (11), we can get another representation of β₂² 

   (12)

From (9) and (12), we get the exact solution of β₁ which is a function of r, β and θ

In the region D⁽¹⁾ , 

    (13)

the solution is a single valued function

     (14)

where

           

                                                                                                                                                                   (15)

           

    (16)

and

          

   (17}

And in the region D⁽²⁾ :

 (18)

the solution is a double valued function, one branch is 

     (19_1)

which is in the same form as (14),

and the other branch is

    (19_2)

Furthermore, from the equality (6), we obtain the exact solution for β₂,

In the region  D⁽¹⁾, the solution is also a single valued function of r, β and θ

  (20)

And in the region D⁽²⁾ , the solution is also a double valued function, one branch is

   (21_1)

and the other branch is 

   (21_2)

It can be known from equation (3) that when the solutions of β₁ and β₂ of the collision problem are obtained, the solutions of v₁, v₂, P_1a, P_2a, K_2a are also obtained naturally. For examples,

  

and

Based on these fundamental quantities, we can introduce three new quantities to describe the consequences of collisions from a mechanical point of view.

1. For the impacting particle 1 ,the scattering coefficient R_S which is the ratio of the magnitude of momentum P_1a to the magnitude of the momentum P_1b, i.e.

   (22)

   2. For the impacted particle 2, the absorption momentum coefficient R_AM which is the the ratio of the magnititude of momentum P_2a to the magnititude of the momentum P_1b, i.e.

    (23)

   3. For the impacted particle 2, the absorption kinetic coefficient R_AK  which is the the ratio of  the kinetic energy K_2a to the kinetic energy K_1b, i.e.

    (24)

More precisely: these ratios should take different forms in regions D⁽¹⁾ and D⁽²⁾ respectively.

In the region D⁽¹⁾ , these ratios are single valued respectively

   (22.1)

   (23.1)

 (24.1)

In the region D⁽¹⁾ , these ratios are double valued respectively

    (22.2_1)

    (22.2_2)

(23.2_1)

(23.2_2)

  (24.2_1)

  (24.2_2)

III.  Understand the complex expressions obtained 

Obviously, all the exact expressions obtained in the previous section are very complex and difficult to understand directly and quickly. However, they should imply some important deep rules that are useful for studying high-speed elastic collisions. Therefore, understanding these complex precise expressions remains highly desirable and necessary.

In fact, with the help of mathematical system software, through precise formula derivation, high-precision numerical calculation and graphic analysis, many important results have been achieved in understanding the precise expression obtained in this paper.

    1.  It is proved that the mediation function satisfies that

and that

This fact means that, at the edge sinθ = 1/r of the region D⁽²⁾, for each mechanical quantity, the two branches of the double valued function must coincide with each other. This allows two branches of the same mechanical quantity to be considered as a whole.

    2.  Through graphical analysis, the common properties of the elastic collision of two particles with non-zero static mass in classical mechanics and relativistic mechanics can be visually checked. 

In the classical mechanics, the scattering coefficient R_S of the particle 1, the absorption momentum coefficient R_AM and absorption kinetic energy coefficient R_AK  of the particle 2 depend on only two variables r and θ. Similar to the case of relativity, they also have different representations in the region  D⁽¹⁾ and  D⁽²⁾ respectively (see my bloq paper mentioned in section I)

In D⁽¹⁾,

 (25.1)

  (26.1)

  (27.1)

and in D⁽²⁾，they are double valued

  (25.2_1)

 

  (25.2_2)

  (26.2_1)

  (26.2_2)

  (27.2_1)

  (27.2_2)

Since all functions from (25.1) to (27.2_2) have two variables r and θ, the value distribiusion of  these functions in regions  D⁽¹⁾ and  D⁽²⁾ can be visualized visually by their function graphs. See below:

In D⁽¹⁾

Fig 2. Classical Scattering Coefficient R_S⁽¹⁾

Fig 3.  Classical Absorption Momentum Coefficient R_AM⁽¹⁾

Fig 4.  Classical Absorption Kinetic Energy Coefficient R_AK⁽¹⁾

In the part of D(2) where 1 < r ≤ 5,

Fig 5.  Classical Scattering Coefficient R_S⁽²⁾

Fig 6.  Classical Absorption Momentum Coefficient R_AM⁽²⁾

Fig 7.  Classical Absorption Kinetic Energy Coefficient R_AK⁽²⁾

These important coefficients of classical mechanics described above can now be compared with their relativistic counterparts in this paper. Since all the corresponding coefficients of relativity have an independent variable β (= u/c) that reflects the collision velocity, this comparison can be made at multiple collision velocity levels. For short, this paper specifically divides the following three collision levels for comparison, namely (a) near-classical high-speed collision (β = 1/1000), (b) ultra-high-speed collision (β = 0.5) and (c) near-light speed collision (β = 0.9).

(a) Near-classical high-speed collision (β = 1/1000):

In D⁽¹⁾ ,

Fig 8.  Scattering Coefficient R_S⁽¹⁾ of Near-Classical Hight Speed collision

Fig 9.  Absorption Momentum Coefficient R_AM⁽¹⁾ of Near-Classical Hight Speed collision

 

Fig 10.  Absorption Kinetic Energy Coefficient R_AK⁽¹⁾ of Near-Classical Hight Speed collision

In the part of D⁽²⁾ where 1 < r ≤ 5,

Fig 11.   Scattering Coefficient R_S⁽²⁾ of Near-Classical Hight Speed collision

Fig 12.   Absorption Momentum Coefficient R_AM⁽²⁾ of Near-Classical Hight Speed collision

Fig 13.    Absorption Kinetic Energy Coefficient R_AK⁽²⁾ of Near-Classical Hight Speed collision

 (b) Ultra-high-speed collision (β = 0.5):

In D⁽¹⁾ ,

Fig 14.    Scattering Coefficient R_S⁽¹⁾ of Ultra-Hight-Speed collision

Fig 15.  Absorption Momentum Coefficient R_AM⁽¹⁾ of Ultra-Hight-Speed collision

Fig 16.    Absorption Kinetic Energy Coefficient R_AK⁽¹⁾ of Ultra-Hight-Speed collision

In the part of D⁽²⁾ where 1 < r ≤ 5,

Fig 17.    Scattering Coefficient R_S⁽²⁾ of Ultra-Hight-Speed collision

Fig 18.  Absorption Momentum Coefficient R_AM⁽²⁾ of Ultra-Hight-Speed collision

Fig 19.    Absorption Kinetic Energy Coefficient R_AK⁽²⁾ of Ultra-Hight-Speed collision

(c) Near-light speed collision (β = 0.9) :

In D⁽¹⁾ ,

Fig 20.    Scattering Coefficient R_S⁽¹⁾ of Near-Hight Speed collision

Fig 21.  Absorption Momentum Coefficient R_AM⁽¹⁾ of Near-Light Speed collision

Fig 22.    Absorption Kinetic Energy Coefficient R_AK⁽¹⁾ of Near-Light Speed collision

In the part of D⁽²⁾ where 1 < r ≤ 5 ,

Fig 23.    Scattering Coefficient R_S⁽²⁾ of Near-Hight Speed collision

Fig 24.  Absorption Momentum Coefficient R_AM⁽²⁾ of Near-Light Speed collision

Fig 25.    Absorption Kinetic Energy Coefficient R_AK⁽²⁾ of Near-Light Speed collision

By the comparison, one can find out the following common properties of this kind of elastic collisions of two particles between the classical mechanics and the relativistic mechanics:

Common properties 1. The mass ratio r has a common critical value, r = 1, which divides the independent variable region of these coefficients into two parts D⁽¹⁾ and D⁽²⁾. In region D⁽¹⁾, when θ is fixed and r increases, R_S⁽¹⁾ and R_AM⁽¹⁾ decrease monotonically, while R_AK⁽¹⁾ increases monotonically.

When r is fixed and θ increases, R_S⁽¹⁾ decreases monotonically, while R_AM⁽¹⁾ and R_AK⁽¹⁾ both increase monotonically. If θ = π, R_AM⁽¹⁾ and R_AK⁽¹⁾ reach their possible maximum values, and R_AM⁽¹⁾'s maximum value is even greater than 1.

Especially in a small region around the specific parameter point r =1 and θ = π, regardless of the magnitude of the impact velocity, the two coefficients R_S⁽¹⁾ and R_AM⁽¹⁾ almost reach their maximum values respectively, that is, R_S⁽¹⁾ almost  reachs 1,  R_AM⁽¹⁾ almost reachs 2, and the coefficient R_AK⁽¹⁾ almost reaches its minimun value, which is zero.  This is a special case that the author is most concerned about, because in this case, the reflection of the impacting particle 1 is the strongest, and the momentum absorbed by the impacting particle 2 from the impacting particle 1 is as high as 2 times the original impact momentum, but the particle 2 does not absorb the kinetic energy of the collision. This implies an odd situation where the struck particle 2 is not affected by the most intense collision from particle 1, although this situation is usually theoretically used in the derivation of light pressure^[2],[3] or air pressure^[3],[4] formulas.

Common properties 2. When r = 1, regardless of the collision speed, the three coefficients R_S⁽¹⁾, R_AM⁽¹⁾ and R_AK⁽¹⁾ all indicate that the scattering angle θ of the impacting particle 1 can only be an acute angle, otherwise particle 1 will stop moving immediately upon collision, and transfer all its momentum and kinetic energy to the impacted particle 2. The figures below show how these coefficients depend on the scattering angle θ (0 ≦ θ ≦ π) and the impact velocity. The blue curve refers to the situation in classical mechanics, and the red, purple, and green curves refer to the near-classical high-speed, ultra-high-speed, and near-light-speed situations, respectively.

Fig 26. Distribution of R_S⁽¹⁾ with respect to scattering angle θ and different impact velocity levels.

Fig 27. Distribution of R_AM⁽¹⁾ with respect to scattering angle θ and different impact velocity levels.

Fig 28. Distribution of R_AK⁽¹⁾ with respect to scattering angle θ and different impact velocity levels.

Common properties 3. In region D⁽²⁾, the scattering angle θ must be less than or equal to arcsin(1/r) for any r ( > 1). If θ < arcsin(1/r), all three coefficients are double-valued. If θ = arcsin(1/r), the two branches of each coefficient coincide with each other. Also for any r (>1), the maximum and minimum values are reached at theta = 0 for all three coefficients.

The authors point out that it can be shown that in this region, if θ = 0, the maxima of R_AM⁽²⁾ and R_AK⁽²⁾ both approach zero as r → ∞, regardless of the magnitude of the impact velocity.

    3.  The above analysis points out that in the elastic collision of two particles with non-zero rest mass, the results of classical mechanics are very consistent or very close to the results of special relativity mechanics in most cases. So what are the main differences between the two mechanics? As we all know, the special theory of relativity is based on the fact that the speed of light is constant, and the motion speed of ordinary matter with non-zero static mass will not reach the speed of light. Therefore, the main difference between the two mechanics in elastic collision should be evident when the collision speed approaches the speed of light.

The author did find an important difference based on the exact solution obtained. Consider the velocity v₂^(2_2)(r,β,θ) of the impacted particle 2 after the collision in D⁽²⁾. Although the authors have just pointed out that when θ = 0, the maxima of R_AM⁽²⁾ and R_AK⁽²⁾ both approach zero as r → ∞, regardless of the magnitude of the impact velocity,  it is interesting that this velocity v₂^(2_2) should not tend to zero and should exceed the impact velocity u. Therefore, when r > 1， the rate

   (28)

can reasonably be called the overspeed coefficient of the impacted particle 2.  This coefficient depends obviously on r and β, while it is independent of β in the classical mechanics. Figure 29 shows the clear difference between the relative case and the classical case for the coefficients R_overS(r,β), where the upper surface is the classical case and the lower surface is the relativistic case. Figure 29 shows the clear difference between the relative case and the classical case for the coefficients R_overS(r,β), where the upper surface is the classical one and the lower surface is the relativistic one.

Fig 29. The clear difference between relativistic case and classical case

Furthermore, it is proved that, for given β, if let r → ∞, then R_overS(r,β) has a limit

    (29)

while this limit is 1 in the classical case. 

IV.    Elastic collision of a light-speed particle with an ordinary particle

In the previous discussion, it was assumed that both particles in the collision had non-zero masses. This assumption loses the important collision possibility that the impacting particle 1 is a light-speed particle. In fact, in the framework of relativity and quantum optics, the well-known Compton scattering^[5] has taken into account the possibility of this loss.

However, there is a question about Compton scattering, whether this collision between a given photon and a given electron can be considered a purely elastic collision of these two particles because of the frequency (or the wavelength of the photon before the collision)) is different from the frequency (or wavelength) of the photon after the collision. According to quantum theory, the photons before and after the collision should not be the same quantum. Moreover, the frequency (or wavelength) change process of a given photon in the collision should be a complex interaction process between the photon's electromagnetic field and the charged particle. If this process is viewed as a simple elastic collision between a photon and an electron, it raises a series of deeper and more complex problems that need to be further considered. The authors believe the trouble comes from the uncertainty caused by the wave-particle duality in quantum mechanics.

Therefore, in order to avoid the above troubles, this paper deliberately ignores whether particle 1 is a quantum, whether there is electromagnetic interaction or other interaction in the collision when studying the collision of the light-speed-particle 1 with the ordinary particle 2 with static mass, and only regards the collision as a is a purely elastic collision in special relativistic mechanics.  In this way, according to the special theory of relativity, the particle 1 with light-speed can only have energy E_1b and momentum P_1b  before the collision, and only energy E_1a and momentum P_1a  after the collision. And assume that the impacted particle 2 with static mass m₂ is still in a static state before the collision, and the velocity w (a vector) is obtained after the collision. Before collision, the energy and momentum of the particle 2 are E_2a (= m₂c² ) and P_2a (= 0) respectively, and after the collision, its energy and momentum are respectively

   (30)

and

   (31)

The three non-zero vectors P_1b, P_1a and P_2a are still distributed according to the Fig 1. 

According to the special relativity,  for the particle 1 of light-speed

    (32)

and

  (33)

The laws of conservation of energy and momentum mean that

  (34)

and that

   (35)

Giving the quantities E1a, m2 and the scattering angle θ, the key unknown quantities w and E1b can be solved from equations (32), (33), (34) and (35) by a similar method used in Section II. In order to save space, the solution process is omitted, and w and E1b are directly given as follows

 (36)

   (37)

where

   (38)

The parameter r_o is similar to the parameter r defined by (8). The equality (37) is equivalent to the well-known Compton Scattering formula^[5]

(39)

if the wavelength of matter introduced by de Broglie^[6],[7]

         (40)

is used for the particle 1 of light-speed.

Using a method similar to Section 2, based on the key solutions (36) and (37), concepts such as the scattering coefficient R_S, the absorption momentum coefficient R_AM, and the absorption kinetic energy coefficient R_AK can be established. Concretely，

   (41)

   (42)

and

   (43)

These three coefficients can also be represented visually by their function graphs, respectively.

Fig 30.

Fig 31.

θ

Fig 32. 

It is not difficult to see that for all values of the independent variables ro and θ (0 ≦ r_o, 0 ≦ θ < π), these three coefficients are single-valued. Therefore, the main difference between the elastic collision of tachyon particles and ordinary particles and the collision of two ordinary particles is that the three important coefficients of the former are all single-valued, and the corresponding coefficients of the latter all have a double-valued region D⁽²⁾.  Interestingly, if the regions of ro and θ

   (44)

considered as the region D⁽¹⁾ of r and θ, then the three important coefficients in different cases are correspondingly similar in quality.  In particular, when r_o → 0 and θ → π or when r → 0 and θ → π, R_AM always approaches 2 and R_AK always approaches 0 in any case. Although in this special case with a small range of parameters, the impacted particle 2 cannot absorb any kinetic energy and can hardly move, the fact that in this small region the scattering coefficient Rs ≈ 1 and the absorption momentum coefficient R_AM ≈ 2 is often used for the derivations of the formulas for light pressure and air pressure^[3],[4],[5]，and is also used in the design of the material on the side of the solar sail facing the sun.  

In general, the kinetic energy absorption efficiency of the impacted particle 2 increases significantly if a larger r or larger r_o is chosen. For example, if r =0.1 and v =3km (i.e., β =0.00001) in a relativistic elastic collision, then

If let r_o = 0.1, then

In addition, if the collision is not purely elastic, the impacted object may absorb more energy, and this part of the energy may be further converted into kinetic energy (such as photoelectric effect, etc.) to promote the movement of the impacted object, but this part is beyond the scope of this paper range.

V.    Conclusion

Because the exact solution obtained in this paper is the simplest and most basic case of the interaction of two particles, the authors believe that these results can be used for the exploration of more complex interactions between matter, especially for the study of high-speed particle interactions.

Before concluding this article, the author would like readers to note that Mathematica, a mathematical system software developed by the Wolfram Research Institute, has become an indispensable assistant in his research.

References

[1] L. D. Landau and E. M. Lifschitz, MECHANICS, THIRD EDDITION, 1976,  Butterworth-Heinenann Linacre House Jordan Hill, Oxford OX2 8DP, ISBN 0 7506 2896 0

[2] P. Lebedew: Untersuchen uber die druckkrafte des Lichtes, Annalen der Physik 46, 432 (1901). 

[3] Ludwig Eduard Boltzmann . Ableitung des Stefan schen Gesetzes betreffend die Abhängigkeit der Wärmestrahlung von der Temperatur aus der electromagnetischen Lichttheorie, pp. 291-294 in: Annalen der Physik und Chemie, Band 22, Heft 2, No. 6, 1884

[4] Krönig, A. (1856), "Grundzüge einer Theorie der Gas“, Annalen der Physik 99 (10): 315–322,

[5] Arthur S. Eddington, The internal constitution of the stars, Cambridge University Press, Cambridge New York New Rochelle Melbourne Sydney 1988 ISBN 0 521 33708 9

[6] de Broglie, L. (1923). "Waves and quanta". Nature. 112 (2815): 540

[7] De Broglie, Louis (1925). "Recherches sur la théorie des Quanta". Annales de Physique (in French). 10 (3): 33. 

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