$6.(3$ marks$)$ Particles $\backslash ($ A $\backslash )$ and $\backslash ($ B $\backslash )$ are moving rectilinearly along the same line. At some point, they collide with each other. Their masses and velocities before the collision are shown in the graph. Assume no external force or mass exchange. Determine:

$(1)$ The velocity of the center of mass $\backslash (($G$)\backslash )$ of the system consisting of $\backslash ($ A $\backslash )$ and $\backslash ($ B $\backslash ),$ and the velocities of $\backslash ($ A $\backslash )$ and $\backslash ($ B $\backslash )$ in the center$-$of$-$mass frame of reference before the collision.

P$.3$ of $4$

$(2)$ The linear momentum and the kinetic energy associated with $\backslash ($ G $\backslash )$ before and after the collision.

$(3)$ The total kinetic energy of the system $\backslash (\backslash$left$($E$\_\{$k$\}\backslash$right$)\backslash ),$ and the kinetic energy of the system relative to its center of mass $\backslash (\backslash$left$($E$\_\{$k G$\}\backslash$right$)\backslash ),$ before the collision.

$(4)\backslash ($ E$\_\{$k$\}\backslash )$ and $\backslash ($ E$\_\{$k G$\}\backslash )$ after the collision if $\backslash ($ e$=0,0\mathrm{.}5\backslash )$ or $1.$

$(5)$ The maximum velocity particle $\backslash ($ B $\backslash )$ can have after the collision. $($Specify the frame of reference you use$)$