$2.$ A bullet of mass $\backslash ($ m $\backslash )$ is fired into a block of mass $\backslash ($ M $\backslash )$ that is at rest. The block, with the bullet embedded, slides a distance $\backslash ($ d $\backslash )$ across a horizontal surface before coming to rest. The coefficient of kinetic friction is $\backslash (\backslash$mu$\_\{$k$\}\backslash )$ between the block and the surface.

i$)$ In terms of the given variables find an expression for the bullet's speed, $\backslash ($ v$\_\{\backslash$text $\{$bullet $\}\}\backslash )$ just before impact with the block.

ii$)$ What is the speed of a $10$ g bullet that, when fired into a $10$ kg stationary wood block, causes the block to slide $5\mathrm{.}0$ cm across the surface where $\backslash (\backslash$mu$\_\{$k$\}=0\mathrm{.}3\backslash )?$

iii$)$ How much of the bullet's mechanical energy is transformed into heat during the collision?