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S.B. Karavashkin and O.N. Karavashkina

2. Complete solution of two-body problem in collision theory

To find the additional regularity that will complement the modelling system (8)- (9), conveniently follow the whole way to solve. Suppose, we have available two point masses m₁ and m₂ whose trajectories cross at some point A of the inertial reference frame S (see Fig. 2). The initial locations and speeds of bodies have such values at which the bodies come to the interaction point A simultaneously, which predetermines the interaction at the given point.

Fig. 2. Interaction of two bodies in the inertial reference frame. The red dotted line shows the trajectory of the mass centre of the system

In Fig. 2 we can clearly see the duality of solution that follows from the initial modelling system (8)- (9). As in case when the bodies reflected from each other (continuous lines after the point A) as in case of non-interaction of the point masses (dotted lines after the point A), both laws (8)- (9) remain true.

So the condition complementing the modelling system of equations has first of all to account the bodies reflection at A. And just this stipulates an accent that predestines the interaction at the point A.

To find this condition, we already have available the calculation of trajectories of bodies motion in the mass centre frame. In particular, as it follows from the relation that links the speeds of bodies in the mass centre frame and mutual speed of bodies (21) (which we have revealed in the calculation),

where '₁, '₂ are the bodies momenta in the mass centre frame.

We see from (23) that in this frame, the momenta are generally similar and opposite in sign.

Then it is easy to show that in bodies motion, the momenta '₁ and  '₂ are directed in one line going through the instantaneous location of the mass centre of the system. To do so, let us follow the trajectories of bodies, for example, for last five equal intervals of time before collision (see Fig. 3a).

a                                                                                       b

Fig. 3. Bodies' trajectories during last five equal time intervals before collision: a - in the stationary reference frame S, and b - in the mass centre frame S_m

As we can see from Fig. 3a, the bodies 1 and 2 to come to A simultaneously, they have at each time interval to pass a section of trajectory proportional to the whole length of path from the initial location to A. As it follows from such circumstance that the line connecting both bodies during their motion also moves in parallel to itself, and on this line, in accordance with the theorem of mass centre of the system, the considered mass centre is instantaneously located.

Hence, at each instant of time both bodies in the mass centre frame will move in a common line parallel to the line connecting the bodies in the stationary frame S (see Fig. 3b). Consequently, speeds and momenta of bodies in the mass centre frame will be directed in this line. Furthermore, since by the instant of collision the location of both bodies and location of the mass centre of the system are subtended to the point A of the frame S, in the frame S_m the locations of both bodies are subtended by the collision to the point A' coinciding with the coordinate origin of this frame.