Problem #$3$: $(30$ points$)$ A plug $A$ is released from rest at point $B$ and under the influence of gravity

descends along a smooth circular surface until it embeds itelf in a block $C,$ which $($with the plug

embedded$)$ then slides on a rough horizontal surface until it comes to rest after traveling a distance

$s,$ as shown in Figure $3.$ The radius of the circular surface is $r,$ and the kinetic coefficient of friction

between the block and the horizontal surface is ${}_k,$ the mass of plug $A$ is $m_A,$ the mass of block $C$

is $mc,$ and the downward acceleration due to gravity is $g.($Express all your answers in terms of

these quantities.$)$

a$)$ Determine the speed $v_A$ of the plug when it reaches the bottom of the circular surface but just

before it embeds itself in the block.

b$)$ Determine the speed $v_{AC}$ of the plug A and the block C just after the plug embeds itself in the

block.

c$)$ Draw a free body diagram of the embedded plug$/$block combination as it slides along the

horizontal surface.

d$)$ Find the time $t$ it takes for the plug$/$block combination to come to rest.

e$)$ Find the distance $s$ the plug$/$block combination travels before it comes to rest.