Elastic Collisions in the CM Frame

                As mentioned in Elastic Collisions in the LAB Frame, there are two ways to solve elastic collision problems.  The first of which is in the LAB frame and was demonstrated in the previous post.  In this post, I will demonstrate how to avoid tedious algebra and the need to solve systems of equations by “boosting” into the center of mass (CM) frame.

                The main benefit of boosting into the CM frame is that contributions of linear momentum are eliminated.  When taking the CM frame approach, the process consists of four steps:

     1. Boost into CM frame

     2. Find velocities in CM frame

     3. Boost out of CM frame

     4. Solve for scattering angles in LAB frame using the determined final velocities

To begin, we will consider the same scenario as in the previous post.

As before, we will assume particles of initial masses m₁ and m₂ and initial velocities u₁ and u₂.  While these velocities can be vectors in any direction, we will assume u₁ and u₂ are only along the x axis.  Furthermore, as I previously mentioned, we must know at least one thing about what happens after the collision and will assume that particle 1 scatters at an angle of θ₁.

                First, we will boost into the CM frame.  To do this, we must determine the velocity of the system’s center of mass and then appropriately adjust the initial velocities from the LAB frame such that we observe them from the CM frame.  To determine the velocity of the system’s center of mass, we use

As we boost to the CM frame, the scenario will look a bit different.  Initially, it will appear as though the two particles are approaching the origin, or center of mass, from opposite sides.  Once they collide, the two particles will simply scatter in opposite directions. 

To properly adjust the initial velocities of the particles such that we observe them from the CM frame, we subtract the CM velocity V.  Thus,

and

where u’₁ and u’₂ are the initial velocities of the particles in the CM frame.

 As the particles are always moving in opposite directions, our linear momentum will always cancel, thus we must only worry about the conservation of kinetic energy.  Through algebraic manipulation of our energy and momentum relations, we determine that

and 

Here, v’₁ and v’₂ are the final velocities of the particles within the CM frame.

                Finally, we want to boost back out of the CM frame to determine our scattering angle θ₂.  To do this, we simply add back the velocity V we originally subtracted from each velocity and then use basic trigonometric relations.  Thus,

 and

are used to determine the particle velocities in the LAB frame.  Then, to determine θ₂, we rearrange the equation for the conservation of momentum, determining

Applying this technique for the case in which

 

and

we find that

and

These results are very close to those using the LAB frame procedure.  Most likely, this discrepancy is simply due to rounding error somewhere within our calculations.