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Collision of particles with symmetry due to interaction.

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This question concerns a collision of two identical particles, A and B, each of mass m, which constitute an isolated system that is observed from an inertial frame. Assume that the particles, when sufficiently close, interact only through a potential energy function. At an instant long before the collision (when the distance between A and B is sufficiently large for the potential energy to be neglected) A has position (d, 0, -f) and velocity (0, 0, V) while B has position (-d, 0, f) and velocity (0, 0, -V), where you should assume f and V and d are positive constants.

The attachment contains the full question which goes on to ask for a description of the symmetry of the configuration. The momentum vector, total energy and angular momentum of the system.

There is a further question where by I need to determine which of three, given configurations, is the correct one using the principle of relativity and conservation laws.

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The solutions deals with the collision while the particles interacts at large distances by conserving momentum and energy. This also discuss the position during the interaction.

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Question 1

This question concerns a collision of two identical particles, A and B, each of mass m, which constitute an isolated system that is observed from an inertial frame. Assume that the particles, when sufficiently close, interact only through a potential energy function. At an instant long before the collision (when the distance between A and B is sufficiently large for the potential energy to be neglected) A has position (d, 0, -f) and velocity (0, 0, V) while B has position (-d, 0, f) and velocity (0, 0, -V), where you should assume f and V and d are positive constants.

i)a)
Describe the symmetry of this initial configuration.

The y coordinates of the particles are both zero and the velocities are also having components in only z direction, it means that the particles are in x-z plane and will move in the x-z plane only and hence the axes are taken as shown in the figure. Y-axis in into the plane of the paper, makes it a right handed system.

At the initial moment the positions of the particles are indicated and the direction of velocities (magnitude is V) are indicated as follows.

As it is clear from the figure, initially the particles are equidistance from the origin O and

OA = OB = (d2 + f2)

And as the components d and f are equal in magnitude the AB will be a straight line.

Thus the system of the two particles is always symmetrical about y axis and as the velocity of both is equal in magnitude and the direction is such that both having equal components towards the y axis (along AB) so approaching to the y axis with the same speed so the ...

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