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Physics LibreTexts

15.7: Two-Dimensional Collisions in Center-of-Mass Reference Frame

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Two-Dimensional Collision in Center-of-Mass Reference Frame

Consider the elastic collision between two particles in the laboratory reference frame (Figure 15.9). Particle 1 of mass m1 is initially moving with velocity v1,i and elastically collides with a particle 2 of mass m2 that is initially at rest. After the collision, the particle 1 moves with velocity v1,f and particle 2 moves with velocity v2,f. In section 15.7.1 we determined how to find v1,f, v2,f, and θ2,f in terms of v1,i and θ2,f. We shall now analyze the collision in the center-of-mass reference frame, which is boosted form the laboratory frame by the velocity of center-of-mass given by vcm=m1v1,im1+m2

Because we assumed that there are no external forces acting on the system, the center-of-mass velocity remains constant during the interaction.

clipboard_e29b207c929dcd5e40d63c6905b373696.png
Figure 15.13 Two-dimensional elastic collision in center-of-mass reference frame

Recall the velocities of particles 1 and 2 in the center-of-mass frame are given by (Equation,(15.2.9) and (15.2.10)). In the center-of-mass reference frame the velocities of the two incoming particles are in opposite directions, as are the velocities of the two outgoing particles after the collision (Figure 15.13). The angle Θcm between the incoming and outgoing velocities is called the center-of-mass scattering angle.

Scattering in the Center-of-Mass Reference Frame

Consider a collision between particle 1 of mass m1 and velocity v1,i and particle 2 of mass m2 at rest in the laboratory frame. Particle 1 is scattered elastically through a scattering angle Θ in the center-of-mass frame. The center-of-mass velocity is given by vcm=m1v1,im1+m2

In the center-of-mass frame, the momentum of the system of two particles is zero 0=m1v1,i+m2v2,i=m1v1,f+m2v2,f Therefore v1,i=m2m1v2,i v1,f=m2m1v2,f The energy condition in the center-of-mass frame is 12m1v21,i+12m2v22,i=12m1v21,f+12m2v22,f Substituting Equations (15.7.4) and (15.7.5) into Equation (15.7.6) yields v1,i=v1,f (we are only considering magnitudes). Therefore v2,i=v2,f Because the magnitude of the velocity of a particle in the center-of-mass reference frame is proportional to the relative velocity of the two particles, Equations (15.7.7) and (15.7.8) imply that the magnitude of the relative velocity also does not change |v1,2,i|=|v1,2,f| verifying our earlier result that for an elastic collision the relative speed remains the same, (Equation (15.2.20)). However the direction of the relative velocity is rotated by the center-of-mass scattering angle Θcm. This generalizes the energy-momentum principle to two dimensions. Recall that the relative velocity is independent of the reference frame, v1,iv2,i=v1,iv2,i In the laboratory reference frame v2,i=0, hence the initial relative velocity is v1,2,i=v1,2,i=v1,i, and the velocities in the center-of-mass frame of the particles are then V1,i=μm1v1,i v2,i=μm2v1,i Therefore the magnitudes of the final velocities in the center-of-mass frame are v1,f=v1,i=μm1v1,2,i=μm1v1,2,i=μm1v1,i v2,f=v2,i=μm2v1,2,i=μm2v1,2,i=μm2v1,i

Example 15.8 Scattering in the Lab and CM Frames

Particle 1 of mass m1 and velocity v1,i by a particle of mass m2 at rest in the laboratory frame is scattered elastically through a scattering angle Θ in the center of mass frame, (Figure 15.14). Find (i) the scattering angle of the incoming particle in the laboratory frame, (ii) the magnitude of the final velocity of the incoming particle in the laboratory reference frame, and (iii) the fractional loss of kinetic energy of the incoming particle.

clipboard_e3a1e13ed3ee7315497c334947adc7de2.png
Figure 15.14 Scattering in the laboratory and center-of-mass reference frames

Solution

i) In order to determine the center-of-mass scattering angle we use the transformation law for velocities v1,f=v1,fvcm In Figure 15.15 we show the collision in the center-of-mass frame along with the laboratory frame final velocities and scattering angles.

clipboard_e66bbca2b1c75642216f7acf09b1667ce.png
Figure 15.15 Final velocities of colliding particles

Vector decomposition of Equation (15.7.15) yields v1,fcosθ1,i=v1,fcosΘcmvcm v1,fsinθ1,i=v1,fsinΘcm where we choose as our directions the horizontal and vertical Divide Equation (15.7.17) by (15.7.16) yields tanθ1,i=v1,fsinθ1,iv1,fcosθ1,i=v1,fsinΘcmv1,fcosΘcmvcm Because v1,i=v1,f, we can rewrite Equation (15.7.18) as tanθ1,i=v1,isinΘcmv1,icosΘcmvcm

We now substitute Equations (15.7.12) and vcm=m1v1,i/(m1+m2) into Equation (15.7.19) yielding tanθ1,i=m2sinΘcmcosΘcmm1/m2 Thus in the laboratory frame particle 1 scatters by an angle θ1,i=tan1(m2sinΘcmcosΘcmm1/m2)

ii) We can calculate the square of the final velocity in the laboratory frame v1,fv1,f=(v1,f+vcm)(v1,f+vcm) which becomes v21,f=v21,f+2v1,fvcm+v2cm=v21,f+2v1,fvcmcosΘcm+v2cm

We use the fact that v1,f=v1,i=(μ/m1)v1,2,i=(μ/m1)v1,i=(m2/m1+m2)v1,i to rewrite Equation (15.7.23) as v21,f=(m2m1+m2)2v21,i+2m2m1(m1+m2)2v1,icosΘcm+m21(m1+m2)2v21,i Thus v1,f=(m22+2m2m1cosΘcm+m21)1/2m1+m2v1,i

(iii) The fractional change in the kinetic energy of particle 1 in the laboratory frame is given by K1,fK1,iK1,i=v21,fv21,iv21,i=m22+2m2m1cosΘcm+m21(m1+m2)21=2m2m1(cosΘcm1)(m1+m2)2 We can also determine the scattering angle Θcm in the center-of-mass reference frame from the scattering angle θ1,i of particle 1 in the laboratory. We now rewrite the momentum relations as v1,fcosθ1,i+vcm=v1,fcosΘcm v1,fsinθ1,i=v1,fsinΘcm

In a similar fashion to the above argument, we have that tanΘcm=v1,fsinθ1,fv1,fcosθ1,f+vcm Recall from our analysis of the collision in the laboratory frame that if we specify one of the four parameters v1,f,v2,f,θ1,f or v1,f then we can solve for the other three in terms of the initial parameters v1,i and v2,i. With that caveat, we can use Equation (15.7.29) to determine Θcm


This page titled 15.7: Two-Dimensional Collisions in Center-of-Mass Reference Frame is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Peter Dourmashkin (MIT OpenCourseWare) via source content that was edited to the style and standards of the LibreTexts platform.

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