Problem $4$:

$25$ Points

A $350-kg$ motorized cart is moving along straight, frictionless tracks with negligible air resistance. The cart initially has a total mass $($cart and contents$)$ of $350$ kg and is traveling with an initial velocity of $10\mathrm{.}0\frac{m}{s}.$ Find the final velocity of the cart in each of the following cases. Additionally, calculate the change in kinetic energy and center of mass where indicated:

A $50-kg$ mass is thrown out of the cart with a velocity of magnitude $15\mathrm{.}0\frac{m}{s}$ in the same direction as the cart's initial velocity.

A$)$ Find the final velocity of the cart.

B$)$ Find the final velocity of the center of the mass.

C$)$ Calculate the initial and final kinetic energy and the change in kinetic energy.

A $90-kg$ mass is thrown backward out of the cart with a velocity of $20\mathrm{.}0\frac{m}{s}$ relative to the cart's initial motion.

D$)$ Find the final velocity of the cart.

E$)$ Calculate the initial and final kinetic energy and the change in kinetic energy.