A $0\mathrm{.}1-kg$ projectile is launched into a $10-kg$ block, initially at rest with an uncompressed spring secured at both ends.

$($a$)$ Draw a free body diagram of the block after it was struck by the bullet.

$($b$)$ Derive the differential equation of motion of the system.

$($c$)$ Write the solution to the differential equation of motion.

$($d$)$ Determine the springs maximum compression.