A $0\mathrm{.}500$ kg sphere moving with a velocity given by

$(2\mathrm{.}003\mathrm{.}50+1\mathrm{.}00)$ m$/$s

strikes another sphere of mass $1\mathrm{.}50$ kgmoving with an initial velocity of

$(1\mathrm{.}00+2\mathrm{.}002\mathrm{.}80)$ m$/$s$.$

$($a$)$ The velocity of the $0\mathrm{.}500$ kg sphere after the collision is

$(0\mathrm{.}50+3\mathrm{.}008\mathrm{.}00)$ m$/$s$.$

Find the final velocity of the $1\mathrm{.}50$ kg sphere.

$=$

$(0\mathrm{.}167)$i$+(0\mathrm{.}167$j$)+(0\mathrm{.}200)$k

m$/$s

Identify the kind of collision $($elastic$,$ inelastic, or perfectly inelastic$).$

elasticinelastic$$ perfectly inelastic

$($b$)$ Now assume the velocity of the $0\mathrm{.}500$ kg sphere after the collision is

$(0\mathrm{.}250+0\mathrm{.}6251\mathrm{.}85)$ m$/$s$.$

Find the final velocity of the $1\mathrm{.}50$ kg sphere.

$=$

$(0\mathrm{.}250)$i$+(0\mathrm{.}625)$j$+(1\mathrm{.}85)$k

m$/$s

Identify the kind of collision.

elasticinelastic$$ perfectly inelastic

$($c$)$ Take the velocity of the $0\mathrm{.}500$ kg sphere after the collision as

$(1\mathrm{.}00+2\mathrm{.}50+$ a$)$ m$/$s$.$

Find the value of aand the velocity of the $1\mathrm{.}50$ kg sphere after an elastic collision. $($Two values of a are possible, a positive value and a negative value. Report each with their corresponding final velocities.$)$

a $($positive value$)$v$2$f $=$ m$/$s a $($negative value$)$v$2$f $=$ m$/$s