SELF

58

S.B. Karavashkin and O.N. Karavashkina

The problem takes another way in the collision theory. It is supposed in frames of this theory that masses interact during infinitesimal time interval of immediate mass contacting. Out of the moment of contacting, masses move inertially and independently of each other. With it, "as the collision forces are very large, during the collision time not just collision forces but their momenta are considered as the measure of interaction. The collision momentum

where F_e^av, is the average force of collision, is finite. The momenta of non-collision forces during the time will be very small and we may neglect them. Further we will denote the speed of point in the beginning of collision , and the speed in the end of collision . Then the theorem of variation of momentum of the point in the collision will take the following form:

i.e., the variation of momentum of a material point during the collision is equal to the sum of collision momenta affecting the point" [5, p. 412]. "The equation (7) is the fundamental equation of collision theory and takes the same part in collision theory as the fundamental law of dynamics m = in studying the motions under the non-colliding forces affection" [ibidem].

The analogy shown by S.M. Targ in the fundamental equations of collision theory and scattering theory enables us to use the common approaches to solving the problems related to the transition to the frame of mass centre of the system.

Noting the features of statement of the problem in collision theory, the modelling system of equations in the two-body problem will consist in general case of two equations [6, p. 61]:

equation of the energy conservation

and equation of the momentum conservation

The system (8)- (9) represented in general form has a direct solution only in 1 D case of problem, when we may write (9) in scalar form. In case when masses before collision move under some angle to each other, the system (8)- (9) has not a direct solution, as we have only three modelling equations for four sought projections of speeds after collision  u_1x , u_1y , u_2x , u_2y .

In scattering theory this difficulty is avoided through passing to the frame of two-bodies mass centre. In that technique, the additional considerations are introduced in which the researchers proceed from the condition that the system under consideration is conservative. This condition is common for scattering theory and collision theory, which allows to use this technique equally in both conceptual approaches. As this technique will be important for our study, let us briefly recall it.