Problem $6(8$ pts$.)$

A spring with spring constant k is attached to a stationary mass $2$m$,$ as shown in Fig. $6\mathrm{.}12.$

A mass m travels to the right and sticks to the left end of the spring and compresses the

spring. The coefficient of friction $($both static and kinetic$)$ between the masses and the

ground is mu$.$ Let v be the speed of m right before it hits the spring.

$($a$)$ What is the largest value of v for which the mass $2$m never moves? $($Don$\text{'}$t forget that

m feels the friction too.$)$ Assume that the spring is sufficiently long so that m doesn't

collide with $2$m$.$

$($b$)$ Assume that v takes on this largest value. The spring will compress and then expand

back out and push m to the left. What will m $\text{'}$s speed be by the time the spring

expands back to its relaxed length? $($Again$,$ don't forget that m feels the friction.$)$