You are designing a rolling platform that will use cannon balls $($slugs$)$ as a means of propulsion. Your rolling platform starts from rest has a total initial mass of $m_{\text{i}\text{n}\text{i}\text{t}\text{i}\text{a}\text{l}}($including the three slugs onboard$).$ Each slug has a mass of $m_{\text{s}\text{l}\text{u}\text{g}\text{,}}$ and leaves your platform with a velocity $($relative to the platform$)$ of $v_{\text{s}\text{l}\text{u}\text{g}}$ and an angle of ${}_{\text{s}\text{l}\text{u}\text{g}}$ as shown in the figure. Assume there is no friction in the horizontal direction $($perfect wheel bearings$)$

Calculate:

Calculate the final velocity of your platform $(v_{\text{v}\text{e}\text{h}\text{i}\text{c}\text{l}\text{e}})$ after you have separately fired each of the three slugs.

Intermediate Steps:

a$.$ Use conservation of momentum for a system of particles to calculate the velocity of the platform after the first slug has been fired.

Remember that the platform mass will be reduced by $m_{\text{s}\text{l}\text{u}\text{g}}$ after firing.

b$.$ The final velocity from step a will become the initial velocity used to simulate the second slug shot. The velocity of the second slug will still be $v_{\text{s}\text{l}\text{u}\text{g}}$ but that velocity is relative to the platform, not the ground. To calculate momentum change you will need to know the velocity of the slug relative to the $($non$-$moving$)$ ground.

c$.$ Use conservation of momentum again to calculate the velocity of the platform after the second shot has been fired.

d$.$ Repeat step b&c one more time to find the velocity of the platform after the third slug has been fired.

SOLVE IN GENERAL TERMS