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Last modified: January 26, 2018

Center of Mass Frame
--------------------

We can always choose a frame of reference where the center of 
mass is stationary.  Under these circumstances Σp = 0.  Consider, 
the following inelastic collision between 2 masses moving in the 
same direction.

Before:

->v1       ->v2  
m1         m2   

After (masses stick together):

->v3
m1 + m2

In the laboratory frame:

Momentum:  m1v1 + m2v2 = (m1 + m2)v3

Energy:  (1/2)m1v12 + (1/2)m2v22 = (1/2)(m1 + m2)v32

In the COM frame:

Momentum:  m1(v1 - v3) + m2(v2 - v3) = 0

Energy:  (1/2)m1(v1 - v3)2 + (1/2)m2(v2 - v3)2 = (1/2)(m1 + m2)v32

Add some numbers

m1 = 1, v1 = 4
m2 = 2, v2 = 2

Laboratory frame:

Momentum: 1*4 + 2*2 = 3*v3  

          => v3 = 8/3 = 2.67

Energy: (1/2)*1*16 + (1/2)*2*4 ≠ (1/2)*3*64/9 

         => 1.35 units of energy lost in collision.

COM frame:

Momentum: 1*(4 - 2.67) + 2*(2 - 2.67) 

          => 0

Energy: (1/2)*1*(4 - 2.67)2 + (1/2)*2*(2 - 2.67)2 

         => 1.35 units of energy lost in collision.

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