$2.$ You are the ship$$s engineer working for the dread pirate captain Piarrrr Squared. Your ship has cannon of mass M which fire cannonballs of mass m$.$ The deck of the ship is a height h above the water, and when the cannon are fired, they launch a cannonball horizontally at some speed vb$.($You$$ve never been able to measure it$.)$ They fly out to sea and splash in the ocean a distance d from the base of the ship. When the cannon are fired, the recoil causes them to slide backwards on the deck; the ship$$s crew has to push them back into place. You decide you want to fix this, so you decide to put heavy springs behind the cannon. When the cannon are fired, they slide back and compress the spring, which pushes on them and brings them to rest. You need to design this system and need to determine the relationship between the spring constant k$,$ the maximum compression amount $$x$,$ and the other things you have measured $$ the masses m and M$,$ the height of the boat h$,$ and the distance the cannonballs travel d$.($a$)$ What principle of physics causes the cannon to recoil backwards when they fire cannonballs forwards? $($b$)$ Write an equation that relates the backwards velocity of the cannon vc to the forward velocity of the cannonballs vb based on the principle you identified. $($This will also involve m and M$.)($c$)$ What technique will let you relate the forward velocity of the cannonball vb to the height of the boat h and the distance the cannonballs fly d$?($d$)$ Write an equation that relates vb$,$ h$,$ and d$.($e$)$ What principle or technique will let you relate the recoil velocity of the cannon vc to the spring constant k and the distance that the spring is compressed $$x$?($f$)$ Write an equation that relates vc to k and $$x$.($g$)$ Putting together the three equations you$$ve written, relate k and $$x $($the things you need to design about your spring system$)$ to d$,$ h$,$ m$,$ M$,$ and g$.($h$)$ Discuss the dependence of these quantities on each other. Is it generally what you expect?