A cart moves on a frictionless surface with an initial velocity of $U_0($to the right$)$ and a mass of $M_0($without any fluid$).$ Into the cart, fluid flows from both sides by horizontal jets with a uniform velocity of $V$ and surface area $A.$ All velocities given above are in respect to a constant coordinate system.

a$)$ Derive the equation of motion of the cart.

b$)$ Find an explicit solution for the cart's velocity as a function of time.

c$)$ An internal pump is added to the cart and sucks fluid from the cart through a hole, located at the right side, with the surface area of $S$ and fluid velocity of $v_e$ in respect to the cart. For simplicity, assume $v_{e}SVA($do not neglect $S{v}_e^2).$