Example $1$: Launching a Projectile at a Target from "Root$\_$Finding$\_$Part$2.$pdf$"$

Calculate the minimum vo to achieve a solution. If no roots are returned then there is no solution for the specified conditions. $($Exercising the existing model only$)$

Solve the problem using different bracketing methods $($Brent$\text{'}$s $1$& $2,$ Ridder's, TOMS$748).$ Compare results. $($Will require some programming$)$

Solve for at least different values of $xf$ and $yf,$ and compare results. $($select your own values$).$ Then find the minimum value of the theta delta to achieve two roots.

theta$\_$delta $=0\mathrm{.}01$

angles$\_$deg $=$ np$.$arange $,$ theta$\_$delta $)$

Try solving the problem using an open method for root finding $($i$.$e$.$ Newton or fsolve$).$ Compare results with the bracketing method.

$\backslash$table$[[,,,,,,,,],[$vo$,,,$vo$,,,$vo$,,],[xf,,,xf,,,xf,,],[$yf$,,,$yf$,,,$yf$,,],[g,,,$g$,,,$g$,,],[,,,,,,,,],[,,,,,,,,],[,$root $1,$root $2,,$root $1,$root $2,,$root $1,$root $2],[$bisect$,,,$bisect,,,bisect,,$],[$Brent $1,,,$Brent $1,,,$Brent $1,,],[$Brent $2,,,$Brent $2,,,$Brent $2,,],[$Ridder$,,,$Ridder,,,Ridder,,$],[$Toms$,,,$Toms,,,Toms,,$],[,,,,,,,,],[$Newton$,,,$Newton,,,Newton,,$],[$fsolve$,,,$fsolve,,,fsolve,,$],[,,,,,,,,],[,,,,,,,,]]$