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$0=$r$\_\{2\}$

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The left end of a spring is fixed to a wall and the right end is initially at rest at position $\backslash ($ x$=0\backslash ).$ The spring constant of the spring is $\backslash ($ k $\backslash ).$ Two identical blocks A and B of total mass $2$ m are sliding together across a horizontal surface towards the spring with initial speed th as shown in Figure $1.$ Friction between the surface and the blocks is negligible, and the two blocks are not attached to each other.

As time $\backslash ($ t$-0\backslash )($not shown$),$ the blocks reach the right end of the spring, and Block $\backslash (\backslash$Lambda $\backslash )$ attaches to the spring.

At time $\backslash ($ t$\_\{1\}\backslash )$ the blocks momentarily come to rest and the right end of the spring is at position $\backslash ($ x$=-$x$\_\{\backslash$text $\{$mas $\}\}\backslash )$ as shown in Figure $2.$

At time $\backslash ($ t$\_\{2\}\backslash )$ the blocks are moving to the right, but the right end of the spring has not yet returned to position $\backslash ($ x$=0\backslash )$ as shown in Figure $3.$

At time $\backslash ($ t$\_\{3\}\backslash )($not shown$),$ the right end of the spring, with Block A attached, again reaches $\backslash ($ x$=0\backslash ).$ At time $\backslash ($ t$\_\{4\}\backslash )($not shown$),$ the spring reaches its maximum extension and Block A is momentarily at rest.

$($a$)$ On the dots below, which represent blocks A and B $,$ draw and label the forces $($not components$)$ exerted on each block at time $\backslash ($ t$\_\{2\}\backslash ).$ Forces should be represented by arrows that start on$,$ and point away from, each dot.

Block B

$($b$)$ Derive an expression for $\backslash ($ t$\_\{4\}\backslash )$ in terms of $\backslash ($ m$\_\{2\},$ k $\backslash ),$ and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference booklet.

$($c$)$ On the axes below, sketch a graph of the velocity of Block B as a function of time for $\backslash ($ t $\backslash$geq $0\backslash ).$ Times $\backslash ($ t$\_\{1\}\backslash )$