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The velocity of two objects after collision (V'_1,V'_2)can be calculated from velocity before collisions (V_1,V_2) and mass of two objects (m_1,m_2).

From conservation of momentum
m_1 V_1+m_2 V_2= m_1 V'_1+m_2 V'_2,
and conservation of energy \tfrac{1}{2}m_1V_1^2+\tfrac{1}{2}m_2V_2^2=\tfrac{1}{2}m_1V_1'^2+\tfrac{1}{2}m_2V_2'^2
So m_1 (V_1-V_1')=m_2(V_2'-V_2)
and \tfrac{1}{2}m_1 (V_1^2-V_1'^2)=\tfrac{1}{2}m_2 (V_2'^2-V_2^2), which means \tfrac{1}{2}m_1 (V_1-V_1')(V_1+V_1')=\tfrac{1}{2}m_2 (V_2'-V_2)(V_2'+V_2)
So V_1+V_1'=V_2'+V_2

i.e. The equation need to be solved are
m_1 V_1'+m_2 V_2'= m_1 V_1+m_2V_2 and V_2'-V_1'=V_2-V_1

The result is
V'_1= \frac{m_1-m_2}{m_1+m_2} V_1 +\frac{2m_2}{m_1+m_2}V_2=V_{cm}+\frac{m_2}{m_1+m_2}(V_2-V_1)=2V_{cm}-V_1
and V'_2=\frac{2m_1}{m_1+m_2}V_1+\frac{m_2-m_1}{m_2+m_1}V_2=V_{cm}+\frac{m_1}{m_1+m_2}(V_1-V_2)=2V_{cm}-V_2
where  V_{cm}=\frac{m_1V_1+m_2V_2}{m_1+m_2}

It means that V'_1-V_{cm} = - (V_1-V_{cm}) and V'_2-V_{cm}= - (V_2-V_{cm})
or V'_{1cm}= -V_{1cm} and V'_{1cm}= -V_{1cm} where V'_{1cm}=V'_1-V_{cm} ...etc.
From the point of center of mass coordinate system: both particles bounce back with the same speed (relative to center of mass).