140 lines
5.2 KiB
Markdown
140 lines
5.2 KiB
Markdown
§17. Elastic collisions
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A collision between two particles is said to be elastic if it involves no change
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in their internal state. Accordingly, when the law of conservation of energy
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is applied to such a collision, the internal energy of the particles may be
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neglected.
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The collision is most simply described in a frame of reference in which the
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centre of mass of the two particles is at rest (the C system). As in $16, we
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distinguish by the suffix 0 the values of quantities in that system. The velo-
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cities of the particles before the collision are related to their velocities V1 and
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V2 in the laboratory system by V10 = M2V/(m1+m2), V20 = -m1V/(m1+m2),
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where V = V1-V2; see (13.2).
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Because of the law of conservation of momentum, the momenta of the two
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particles remain equal and opposite after the collision, and are also unchanged
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in magnitude, by the law of conservation of energy. Thus, in the C system
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the collision simply rotates the velocities, which remain opposite in direction
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and unchanged in magnitude. If we denote by no a unit vector in the direc-
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tion of the velocity of the particle M1 after the collision, then the velocities
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of the two particles after the collision (distinguished by primes) are
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V10' m20120/(m1+m2), V20' = -mjono/(m1+m2).
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(17.1)
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§17
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Elastic collisions
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45
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In order to return to the L system, we must add to these expressions the
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velocity V of the centre of mass. The velocities in the L system after the
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collision are therefore
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V1' =
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(17.2)
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V2' =
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No further information about the collision can be obtained from the laws
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of conservation of momentum and energy. The direction of the vector no
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depends on the law of interaction of the particles and on their relative position
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during the collision.
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The results obtained above may be interpreted geometrically. Here it is
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more convenient to use momenta instead of velocities. Multiplying equations
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(17.2) by M1 and M2 respectively, we obtain
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(17.3)
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P2' muno+m2(p1+p2)/(m1+m2)
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where m = m1m2/(m1+m2) is the reduced mass. We draw a circle of radius
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mv and use the construction shown in Fig. 15. If the unit vector no is along
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OC, the vectors AC and CB give the momenta P1' and P2' respectively.
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When p1 and P2 are given, the radius of the circle and the points A and B
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are fixed, but the point C may be anywhere on the circle.
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C
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p'
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no
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P'2
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B
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A
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FIG. 15
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Let us consider in more detail the case where one of the particles (m2, say) is
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at rest before the collision. In that case the distance OB = m2p1/(m1+m2) = mv
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is equal to the radius, i.e. B lies on the circle. The vector AB is equal to the
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momentum P1 of the particle M1 before the collision. The point A lies inside
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or outside the circle, according as M1 < M2 or M1 > M2. The corresponding
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46
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Collisions Between Particles
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§17
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diagrams are shown in Figs. 16a, b. The angles 01 and O2 in these diagrams
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are the angles between the directions of motion after the collision and the
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direction of impact (i.e. of P1). The angle at the centre, denoted by X, which
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gives the direction of no, is the angle through which the direction of motion
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of m1 is turned in the C system. It is evident from the figure that 01 and O2
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can be expressed in terms of X by
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(17.4)
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C
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p'
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P2
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pi
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P2
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0
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max
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10,
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X
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O2
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O2
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B
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B
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A
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0
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A
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Q
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0
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(a) m < m2
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(b) m, m m m
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AB=p : AO/OB= m/m2
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FIG. 16
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We may give also the formulae for the magnitudes of the velocities of the
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two particles after the collision, likewise expressed in terms of X:
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ib
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(17.5)
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The sum A1 + O2 is the angle between the directions of motion of the
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particles after the collision. Evidently 01 + O2 > 1/1 if M1 < M2, and 01+O2 < 1st
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if M1 > M2.
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When the two particles are moving afterwards in the same or in opposite
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directions (head-on collision), we have X=TT, i.e. the point C lies on the
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diameter through A, and is on OA (Fig. 16b ; P1' and P2' in the same direc-
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tion) or on OA produced (Fig. 16a; P1' and P2' in opposite directions).
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In this case the velocities after the collision are
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(17.6)
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This value of V2' has the greatest possible magnitude, and the maximum
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§17
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Elastic collisions
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47
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energy which can be acquired in the collision by a particle originally at rest
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is therefore
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(17.7)
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where E1 = 1M1U12 is the initial energy of the incident particle.
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If M1 < M2, the velocity of M1 after the collision can have any direction.
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If M1 > M2, however, this particle can be deflected only through an angle
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not exceeding Omax from its original direction; this maximum value of A1
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corresponds to the position of C for which AC is a tangent to the circle
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(Fig. 16b). Evidently
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sin Omax = OC|OA = M2/M1.
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(17.8)
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The collision of two particles of equal mass, of which one is initially at
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rest, is especially simple. In this case both B and A lie on the circle (Fig. 17).
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C
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p'
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P2
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Q2
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B
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A
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0
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FIG. 17
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Then
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01=1x,
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A2 = 1(-x),
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(17.9)
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12
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=
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(17.10)
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After the collision the particles move at right angles to each other.
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PROBLEM
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Express the velocity of each particle after a collision between a moving particle (m1) and
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another at rest (m2) in terms of their directions of motion in the L system.
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SOLUTION. From Fig. 16 we have P2' = 20B cos O2 or V2' = 2v(m/m2) cos O2. The momen-
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tum P1' = AC is given by OC2 = AO2tp12-2AO.p cos or
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Hence
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for m1 > M2 the radical may have either sign, but for M2 > M1 it must be taken positive.
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