Chapter 11 Collision Theory Introduction 11.1 Center of Mass Reference Frame Consider two particles of masses m and interacting via some force. 1m2 Figure 11.1 Center of Mass of a system of two interacting particles Choose a coordinate system (Figure 11.2) in which the position vector of body 1 is given by and the position vector of body 2 is given by 1r2r. The relative position of body 1 with respect to body 2 is given by 12 1 2,=−rrr. Figure 11.2 Center of mass coordinate system. During the course of the interaction, body 1 is displaced by 1dr and body 2 is displaced by , so the relative displacement of the two bodies during the interaction is given by . The relative velocity between the particles is 2dr12 1 2,ddd=−rrr 121212 1 2,,ddddt dt dt==−=−rrrvvv. (11.1.1) 1We shall now show that the relative velocity between the two particles is independent of the choice of reference frame. Let be the vector from the origin of frame RS to the origin of reference frame ′S. Denote the position vector of particle i with respect to origin of reference frame S by and similarly, denote the position vector of particle with respect to origin of reference frame iri′S by (Figure 11.4). i′r Figure 11.4 Position vector of particle in two reference frames. thi The position vectors are related by ii′=+rrR. (11.1.2) The relative velocity (call this the boost velocity) between the two reference frames is given by ddt=RV. (11.1.3) Assume the boost velocity between the two reference frames is constant. Then, the relative acceleration between the two reference frames is zero, ddt==VA0. (11.1.4) Suppose the particle in Figure 11.4 is moving; then observers in the different reference frames will measure different velocities. Denote the velocity of particle in frame thithi S by iiddt=vr, and the velocity of the same particle in frame ′S by iiddt′′=vr. Since the derivative of the position is velocity, the velocities of the particles in two different reference frames are related according to ii′=+vvV. (11.1.5) In frame , the relative velocity is S 212 1 2,=−vvv (11.1.6) The relative velocity in reference frame 12′v′S can be determined from using Equation (11.1.5) to express Equation (11.1.6) in terms of the primed quantities, ()()1,212 1 2 121,2′′′′=−= + − + =−=vvvvVvVvvv′ (11.1.7) and is equal to the relative velocity in frame . For a two-particle interaction, the relative velocity between the two vectors is independent of the choice of reference frame. S In Appendix 8.B, we showed that when two particles of masses and interact, the change of kinetic energy between the final state m1m2B and the initial state A due to the interaction force only is equal to (2212BAKvμΔ= −)v) (11.1.8) where is the reduced mass of the two-particle system. (12 1 2/mm m mμ=+(If Equation (11.1.4) did not hold, Equation (11.1.8) would not be valid in all frames.) In Equation (11.1.8), the square of the final relative velocity ()()12B−vvB is given by ()()()()()()()21,2 1 2 1 2BB BBv =−⋅−vv vvB (11.1.9) and the square of the initial relative velocity ()()12A−vvA is given by ()()()()()()()21,2 1 2 1 2AAAAv =−⋅−vv vvA. (11.1.10) By expressing the change of kinetic energy in terms of the relative velocity, a quantity that is independent of the reference frame, the change in kinetic energy is therefore independent of the choice of reference frame. Characterizing Collisions In a collision, the ratio of the magnitudes of the initial and final relative velocities is called the coefficient of restitution and denoted by the symbol , e BAvev=. (11.1.11) 3If the magnitude of the relative velocity does not change during a collision, , then the change in kinetic energy is zero, (Equation e = 1(11.1.8)). Collisions in which there is no change in kinetic energy are called elastic collisions, ΔK=0, elastic collision . (11.1.12) If the magnitude of the final relative velocity is less than the magnitude of the initial relative velocity, e<1, then the change in kinetic energy is negative. Collisions in which the kinetic energy decreases are called inelastic collisions, ΔK<0, inelastic collision . (11.1.13) If the two objects stick together after the collision, then the relative final velocity is zero, . Such collisions are called totally inelastic. The change in kinetic energy can be found from Equation e= 0(11.1.8), 2121211,222AAmmK v v totally inelastic collisionmmμΔ=− =−+. (11.1.14) If the magnitude of the final relative velocity is greater than the magnitude of the initial relative velocity, , then the change in kinetic energy is positive. Collisions in which the kinetic energy increases are called superelastic collisions, e> 1 . (11.1.15) ΔK > 0, superelastic collision 11.2 Worked Examples 11.2.1 Example: Elastic One-Dimensional Collision Consider the elastic collision of two carts along a track; the incident cart 1 has mass m and moves with initial speed . The target cart has mass and is initially at rest, . Immediately after the collision, the incident cart has final speed and the target cart has final speed . Calculate the final velocities of the carts as a function of the initial speed . 1v1,0m2= 2m1v2,0= 0v1, fv2, fv1,0 Solution Draw a “momentum flow” diagram for the objects before (initial state) and after (final state) the collision (Figure 11.5, with a greatly simplified rendering of a “cart”). 4Figure 11.5 Momentum flow diagram for elastic one-dimensional collision There are no external forces acting on the system, so the component of the momentum along the direction of the collision is the same before and after the collision, . (11.2.1) m1v1,0=−m1v1, f+ 2m1v2, f Note that in the above figure and Equation (11.2.1), the incident cart is taken to be moving backwards if . Equation v1,f> 0 (11.2.1) simplifies to . (11.2.2) v1,0=−v1, f+ 2 v2, f The collision is elastic; the kinetic energy is the same before and after the collision, 12m1v1,02=12m1v1, f2+122m1v2, f2, (11.2.3) which simplifies to . (11.2.4) v1,02= v1, f2+ 2v2, f2 There are many ways to manipulate Equations (11.2.2) and (11.2.4) to solve for the final velocities in
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