Newton's law of motion look the same to all observers in inertial frames of reference.
It is equally true that if momentum is conserved in one inertial reference frame, it is conserved in all inertial frames.

This java applet apply the above concept to one dimensional collision problem.
Two circular objects are confined to move in one diminution (between two gray blocks).
Press start button to run the animation.

Click the mouse button to pause. Click it again to resume the animation.

While the animation is suspended:

Press Reset to reset most parameters to default values.

You can select different frame of reference to view the relative motion of all the objects.
       lab is a laboratory inertial frame.
       m_1, m_2and CM are frame of reference with respect to left circular object m_1,  right circular object m_2 and center of mass for m_1 and m_2.

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).


You are welcomed to check out collision in 2 dimension.