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2.7: Center of Mass of a Many-Body System
https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Variational_Principles_in_Classical_Mechanics_(Cline)/02%3A_Review_of_Newtonian_Mechanics/2.07%3A_Center_of_Mass_of_a_Many-Body_System
A finite sized body needs a reference point with respect to which the motion can be described. The center of mass provides this reference point.
15.4: One-Dimensional Collisions Between Two Objects
https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Classical_Mechanics_(Dourmashkin)/15%3A_Collision_Theory/15.04%3A_One-Dimensional_Collisions_Between_Two_Objects
$m_{1}\left(v_{1 x, i}-v_{1 x, f}\right)\left(v_{1 x, i}+v_{1 x, f}\right)=m_{2}\left(v_{2 x, f}-v_{2 x, i}\right)\left(v_{2 x, f}+v_{2 x, i}\right) \nonumber$ \[m_{1} v_{1 x, i}+m_{2} v_{2 x, i}=m... $m_{1}\left(v_{1 x, i}-v_{1 x, f}\right)\left(v_{1 x, i}+v_{1 x, f}\right)=m_{2}\left(v_{2 x, f}-v_{2 x, i}\right)\left(v_{2 x, f}+v_{2 x, i}\right) \nonumber$ $m_{1} v_{1 x, i}+m_{2} v_{2 x, i}=m_{1} v_{1 x, f}+m_{2}\left(v_{1 x, f}+v_{1 x, i}-v_{2 x, i}\right) \nonumber$ $m_{1} v_{1 x, i}+m_{2} v_{2 x, i}=m_{1}\left(v_{2 x, f}-v_{1 x, i}+v_{2 x, i}\right) v_{1 x, f}+m_{2} v_{2 x, f} \nonumber$

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