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| 17 Elastic collisions |
A collision between two particles is said to be elastic if it involves no change in their internal state. Accordingly, when the law of conservation of energy is applied to such a collision, the internal energy of the particles may be neglected.
The collision is most simply described in a frame of reference in which the centre of mass of the two particles is at rest (the C system). As in 16, we distinguish by the suffix 0 the values of quantities in that system. The velocities of the particles before the collision are related to their velocities \v{v}_1 and \v{v}_2 in the laboratory system by \v{v}_{10} = m_2\v{v}/(m_1+m_2), \v{v}_{20} = -m_1\v{v}/(m_1+m_2), where \v{v} = \v{v}_1-\v{v}_2; see 13.2.
Because of the law of conservation of momentum, the momenta of the two particles remain equal and opposite after the collision, and are also unchanged in magnitude, by the law of conservation of energy. Thus, in the C system the collision simply rotates the velocities, which remain opposite in direction and unchanged in magnitude. If we denote by \v{n}_0 a unit vector in the direction of the velocity of the particle m_1 after the collision, then the velocities of the two particles after the collision (distinguished by primes) are
17.1
In order to return to the L system, we must add to these expressions the velocity \v{V} of the centre of mass. The velocities in the L system after the collision are therefore
17.2
No further information about the collision can be obtained from the laws of conservation of momentum and energy. The direction of the vector \v{n}_0 depends on the law of interaction of the particles and on their relative position during the collision.
The results obtained above may be interpreted geometrically. Here it is more convenient to use momenta instead of velocities. Multiplying equations 17.2 by m_1 and m_2 respectively, we obtain
17.3
where m = m_1m_2/(m_1+m_2) is the reduced mass. We draw a circle of radius mv and use the construction shown in fig15. If the unit vector \v{n}_0 is along OC, the vectors AC and CB give the momenta \v{p}_1' and \v{p}_2' respectively. When \v{p}_1 and \v{p}_2 are given, the radius of the circle and the points A and B are fixed, but the point C may be anywhere on the circle.
15
Let us consider in more detail the case where one of the particles (m_2, say) is at rest before the collision. In that case the distance OB = m_2p_1/(m_1+m_2) = mv is equal to the radius, i.e. B lies on the circle. The vector AB is equal to the momentum \v{p}_1 of the particle m_1 before the collision. The point A lies inside or outside the circle, according as m_1\lt m_2 or m_1\gt m_2. The corresponding diagrams are shown in Figs. 16a, b. The angles \theta_1 and \theta_2 in these diagrams are the angles between the directions of motion after the collision and the direction of impact (i.e. of \v{p}_1). The angle at the centre, denoted by \chi, which gives the direction of \v{n}_0, is the angle through which the direction of motion of m_1 is turned in the C system. It is evident from the figure that \theta_1 and \theta_2 can be expressed in terms of \chi by
17.4
16
We may give also the formulae for the magnitudes of the velocities of the two particles after the collision, likewise expressed in terms of \chi:
17.5
The sum \theta_1 + \theta_2 is the angle between the directions of motion of the particles after the collision. Evidently \theta_1 + \theta_2 > \mfrac{1}{2}\pi if m_1\lt m_2, and \theta_1+\theta_2\lt\mfrac{1}{2}\pi if m_1\gt m_2.
When the two particles are moving afterwards in the same or in opposite directions (head-on collision), we have \chi=\pi, i.e. the point C lies on the diameter through A, and is on OA (Fig. 16b ; \v{p}_1' and \v{p}_2' in the same direction) or on OA produced (Fig. 16a; \v{p}_1' and \v{p}_2' in opposite directions).
In this case the velocities after the collision are
17.6
This value of \v{v}_2' has the greatest possible magnitude, and the maximum energy which can be acquired in the collision by a particle originally at rest is therefore
17.7
where E_1 = \mfrac{1}{2}m_1{v_1}^2 is the initial energy of the incident particle.
If m_1\lt m_2, the velocity of m_1 after the collision can have any direction. If m_1 > m_2, however, this particle can be deflected only through an angle not exceeding \theta_\max from its original direction; this maximum value of \theta_1 corresponds to the position of C for which AC is a tangent to the circle Fig. 16b. Evidently
17.8
The collision of two particles of equal mass, of which one is initially at rest, is especially simple. In this case both B and A lie on the circle Fig. 17.
17
Then
17.9
17.10
After the collision the particles move at right angles to each other.