$(20$ points$)$ A projectile is fired from the origin $($at $y=0m)$ as shown in the figure above, which only shows the general idea but the not to scale at all. The initial velocity components are $v_{0x}=10\frac{m}{s}$ and $v_{0y}=19\mathrm{.}6\frac{m}{s}.$ The projectile reaches maximum height at point $P,$ then it falls and strikes the ground $(20$ m below$)$ at point Q$.$

$($a$)$ How long $($in time$)$ after launch does it reach point P $?$

$($b$)$ How high is P relative to ground?

$($c$)$ How long after point P $($in time$)$ does it reach ground?

$($d$)$ How far is the horizontal distance of point $Q$ relative to origin?