Problem $1.$ The mass$-$damper system with mass $m$

and damping coefficient $b$ is moving in vertical plane

against friction with kinetic friction coefficient $.$

a$)$ Derive the equation of motion $($EoM$)$ for the speed

$v$ using free body diagram and fundamental law of

dynamics. $($Hint: use $F_f=Nsign(v)$ for the friction

force.$)$

b$)$ With $v(0)=v_0>0$ initial condition $($IC$),$ solve

the EoM analytically for the duration when $v(t)>0.$

c$)$ Explain what happens after $v(t)$ reaches zero.

Sketch the full solution $v(t)$ for $t0.$