A $1\mathrm{.}96$ g marble is fired vertically upward using a spring gun. The spring must be compressed $2\mathrm{.}98$ cm if the marble is to just reach a target $10\mathrm{.}9$ m above the marble's position on the compressed spring. $($a$)$ What is the change AUg in the gravitational potential energy of the marble$-$Earth system during the $10\mathrm{.}9$ m ascent? $($b$)$ What is the change AU$,$ in the elastic potential energy of the spring during its launch of the marble? $($c$)$ What is the spring constant of the spring?