Block, Ramp, and a Frictional Surface $(10$ points$)$: A block is released to slide down a frictionless incline of $15,$ and then it encounters a frictional surface with a coefficient of kinetic friction ${}_k=0\mathrm{.}19.$

a$)$ Write the appropriate

Conservation of Energy equation $($like in the last problem$)$ to find the horizontal distance traveled on the frictional surface before the block stops if the length of the incline $($ d $)$ is $18\mathrm{.}4$ m $.$

Hint: Think of this as one process. The left side of your equation represents all energy present at the very beginning, plus the total non$-$conservative work done on the block by friction throughout the process. The right side represents all the energy left at the very end. You do not need to represent any intermediate stage.

b$)$ Find the position where the block stops on the horizontal frictional surface if the block has a mass of $25$ kg and the length of the incline is $d=18\mathrm{.}4m.($Use energy considerations to solve this problem.$)$

Hint: The block will stop when friction has done enough work $(W_{NC})$ to eliminate its kinetic energy. Use the above equation to find $W_{NC}.$ Then calculate the friction force and use $W_{NC}=f{d}_{2}cos$ to find the distance $d_2$ travelled on the frictional surface.