A $0\mathrm{.}500$ kg sphere moving with a velocity given by $(2\mathrm{.}00$hat$(i)-2\mathrm{.}70$hat$(j)+1\mathrm{.}00$hat$(k))\frac{m}{s}$ strikes another sphere of mass $1\mathrm{.}50$ kg moving with an initial velocity of $(-1\mathrm{.}00$hat$(i)+2\mathrm{.}00$hat$(j)-2\mathrm{.}80$hat$(k))\frac{m}{s}.$

$($a$)$ The velocity of the $0\mathrm{.}500$ kg sphere after the collision is $(-0\mathrm{.}70$hat$(i)+3\mathrm{.}00$hat$(j)-8\mathrm{.}00$hat$(k))\frac{m}{s}.$ Find the final velocity of the $1\mathrm{.}50$ kg sphere.

vec$(v)=(-0\mathrm{.}10i+0\mathrm{.}10j+0\mathrm{.}20k)\frac{m}{s}$

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

elastic

inelastic

perfectly inelastic

$($b$)$ Now assume the velocity of the $0\mathrm{.}500$ kg sphere after the collision is $(-0\mathrm{.}250$hat$(i)+0\mathrm{.}825$hat$(j)-1\mathrm{.}85$hat$(k))\frac{m}{s}.$ Find the final velocity of the $1\mathrm{.}50$ kg sphere.

vec$(v)=\frac{1}{1\mathrm{.}50}(0\mathrm{.}500(v_{1i}-v_{1f})+1\mathrm{.}5n_1,3i)$

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Identify the kind of collision.

elastic

inelastic

perfectly inelastic

$($c$)$ Take the velocity of the $0\mathrm{.}500$ kg sphere after the collision as $(-1\mathrm{.}00$vec$(i)+3\mathrm{.}30$vec$(j)+$avec$(k))\frac{m}{s}.$ Find the value of a and 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$)$

$v_{2f}=$

$$ hat$(k)\frac{m}{s}$

a $($negative value$)v_{2f}=)<mi>(</mi>$