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13.3: Projectile Problem
https://phys.libretexts.org/Courses/Prince_Georges_Community_College/General_Physics_I%3A_Classical_Mechanics/13%3A_Newtons_Method/13.03%3A_Projectile_Problem
$\theta_{n+1}=\theta_{n}-\frac{x_{t} \sin 2 \theta_{n}-2 y_{t} \cos ^{2} \theta_{n}-g x_{t}^{2} / v_{0}^{2}}{2 x_{t} \cos 2 \theta_{n}+2 y_{t} \sin 2 \theta_{n}}$ Then plug this $\theta_{0}$ into ... $\theta_{n+1}=\theta_{n}-\frac{x_{t} \sin 2 \theta_{n}-2 y_{t} \cos ^{2} \theta_{n}-g x_{t}^{2} / v_{0}^{2}}{2 x_{t} \cos 2 \theta_{n}+2 y_{t} \sin 2 \theta_{n}}$ Then plug this $\theta_{0}$ into the right-hand side, which returns $\theta_{1}$ ; for the next iteration, plug this $\theta_{1}$ into the right-hand side, which returns $\theta_{2}$ , etc.

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