Use the position function s$($t$)=-16$ t$^2+$v$\_0$ t$+$s$\_0$ for free$-$falling objects. A ball is thrown straight down from the top of a $500-$foot building with an initial velocity of $-30$ feet per second. $($a$)$ Determine the position and velocity v$($t$)$ functions for the ball. s$($t$)=$ v$($t$)=($b$)$ Determine the average velocity, in feet per second, on the interval $[1,3].$ ft $/$ s $($c$)$ Find the instantaneous velocities, in feet per second, when t$=1$ and t$=3,[$ v$(1)=$ ft $/$ s; v$(3)=$ ft $/$ s $]($d$)$ Find the time, in seconds, required for the ball to reach ground level. $($Round your answer to three decimal places.$)($e$)$ Find the velocity of the ball, in feet per second, at impact. $($Round your answer to one decimal place.$)$