$1\mathrm{.}10.$ Return to the experimental sled problem where the equation of motion was

$x^{}=-k(x^{}{)}^2.$

$($a$)$ Convert the system to two first$-$order equations using variables $x$ and $v$

$(=x^{}).$

$($b$)$ Use the Runge$-$Kutta algorithm to numerically integrate part $($a$),$ with

$x(0)=0m,v(0)=200k\frac{m}{h}.$

$($c$)$ Graph $v$ versus $x$ from part $($b$).$ How does this compare with the exact

solution?