A spring $(80$ N$/$m$)$ has an equilibrium length of $1\mathrm{.}00$m$.$ The spring is compressed to a length of $0\mathrm{.}50$m and a mass of $2\mathrm{.}2$kg is placed at its free end on a slope which makes an angle of $41$ with respect to the horizontal. The spring is then released.

a$)$ if the mass is not attached to the spring and we assume that the slope is frictionless, how far up the slope from the compressed point will the mass move before coming to rest?

b$)$ assume a frictionless slope, write down the equation for the distance d the mass moved in terms of compression length of the spring x$1,$ slope angle t$,$ mass m$,$ spring constant k$,$ and gravitational acceleration g$.$

c$)$ if the kinetic coefficient of friction between the block and the slope is $0\mathrm{.}42,$ how far up the slope from the compressed point will the mass move before coming to rest?