A $10-$kg package drops from chute into a $25-$kg cart with a velocity of

$5$ m$/$s$.$ The cart is initially moving toward package at a speed of $1\mathrm{.}0$ m$/$s

and can roll freely with no friction. Determine:

a$)$ the final velocity of the cart,

b$)$ the impulse exerted by the cart on the package,

c$)$ the fraction of the initial energy lost in the impact