Problem $1$ Conservation Laws:

Instructions: Please indicate the principles that you are going to use to solve the problem by stating it and writing the mathematical statement. Show all your manipulations and in your calculations work with units.

The picture below illustrates the process, a$)$ a spring$-$loaded gun, with spring constant $k_s,$ fires a ball of mass, $m,$ after being compressed a distance, $x,$ into a pendulum of mass, $M,$ and length, $L,$ with an initial velocity vec$(v_0).$ Then, the ball embeds in the pendulum, causing it to swing upward completing a circle with a the smallest possible velocity at the top $(V_f).$

a$.$ In terms of the given quantities and known constants $(m,M,L,k_s,x,$ and $g)$ determine the initial velocity of the ball $v_0.$

b$.$ Determine the velocity of the pendulum vec$(V)$ immediately after the collision, in terms of the given quantities $(m,M,L,k_s,x,$ and $g).$

c$.$ In terms of the given quantities $(m.M,L,k,x,$ and $g),$ determine the expression for the final velocity of the pendulum, vec$(V_f).$

EXTRA credit $(10$ pts$)$

Classify the collision and justify your answer algebraically