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

Calculate the minimum v$0$ 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