Problem #$1$: $(35$ points$)$ At any distance $s$ from the starting line of a road race $($along a straight

road$),$ the speed $v=v(s)$ of a runner $($shown in Figure $1!)$ is given as a function of the distance $s$

traveled by the runner according to the relation:

$v(s)=v_o(1-bs{)}^{1-N}$

where the initial speed $v_o$ as well as parameters $b$ and $N$ are all known constants $($and $ts=0v=v_{0}t(s)ss(t)t{v}_0,bNv(t)t{v}_0,$bNa$(s)s{v}_0,$bNa$(t)t{v}_0,bN{v}_o=8mph,b=0\mathrm{.}05m{i}^{-1}N=0\mathrm{.}4ts=60$