Q$3-(25$ Pts$)$ In an upcoming movie, the heroine Momo Constantine is in front pod of a two$-$pod transport which floats on magnetic rails inside one of Elon Musk's hyperloop tunnels $($i$.$e$.,$ no friction$).$ Momo's pods are at rest when the villain of the movie sends a third pod moving at velocity vhat$()$ towards her with a bomb in it $(v>0$ and all action is along the x $-$axis$).$ The bomb is set to explode when it gets sufficiently close to Momo. Since there is no fuel in any of the pods, the only way she can avoid the bitter end is by pushing the back pod towards the bomb before it gets too close.

Luckily, between the two pods is an industrial$-$grade spring $($see the figure$)$ which is compressed to store a potential energy of $U.$ When released, the spring pushes the pods away from each other, converting the stored potential energy to kinetic energy. Since they are designed to lock upon contact, the back pod and the bomb will move together after they collide. The mass of each pod is $m.$ The bomb also has mass $m.($Momo and the spring have negligible mass$).$

We want to find $U_{\text{m}\text{i}\text{n}},$ the minimum value of $U$ that will save Momo by avoiding contact with the bomb carrying pod. For a full grade, show your work and explain your reasoning in all parts.

$($a$)(5$ pts$)$ What is the velocity of the center$-$of$-$mass $($ vec$(v{)}_{\text{c}\text{o}\text{m}})$ of all three pods after the back pod and the bomb are locked together?

Assume $U=U_{\text{m}\text{i}\text{n}}$ for the rest of the questions.

$($b$)(5$ pts$)$ What is the velocity of the back pod $($ vec$(v{)}_{\text{b}\text{a}\text{c}\text{k}\text{p}\text{o}\text{d}})$ on its way towards the bomb? Express your answer in terms of $U_{\text{m}\text{i}\text{n}}$ and $m.$

$($c$)(5$ pts$)$ What is the relation between the final velocities of Momo's pod $($ vec$(v{)}_{\text{M}\text{o}\text{m}\text{o}})$ and the "bomb & back pod combination" $($ vec$(v{)}_{\frac{b}{\text{\&}}bp})$ when $U=U_{\text{m}\text{i}\text{n}}?$

$($d$)(5$ pts$)$ Calculate $U_{\text{m}\text{i}\text{n}}$ in terms of $m$ and $v.$

$($e$)(5$ pts$)$ What percentage of the initial mechanical energy is lost to heat?