Problem $2$: $($a$)[15$ points$]$ Determine the velocity and position of a particle of mass $m$ that is projected vertically upwards with initial velocity $v_o$ in a uniform gravitational field vec$(g)$ in a medium with a frictional force proportional to the velocity. Let the $y-$axis point upwards.

$($b$)[5$ points$]$ Does this particle reach a terminal velocity? Explain.

$($c$)[5$ points$]$ Determine the time that it takes the particle to reach the maximum height.

$($d$)[10$ points$]$ Derive expressions for the velocity and the position of the particle on its upward trajectory for the case that vec$(g)=0$ and compare your answer with that of part $($a$)($i$.$e$.$ Does it faster reach a maximum height or slower? Does it reach higher or lower? Explain.$).$