We begin by defining a system

\(\theta\) is the angle by which the shooter is aimed, the shooter shoots at \(v_0\), the projectile travels a distance of \(R\).

So, define a function \(R(\theta) = R\).

Hence, the goal of this project is to find local mix, min points (critical points that arn't inflection points), which means — at a minimum…

vs

\begin{equation} \frac{dR}{d\theta} = 0 \end{equation}which would therefore indicate a \(\theta\) such that the distance would be the longest.

Hence, to get the longest distance, solve.

There was apparently my old notes on this. But not sure if its helpful.

\begin{align} &y(t), y_0=0, y_f=0 \\ &x(t), x_0=0, y_f=R \\ y(t) =& \frac{-1}{2} gt^2 + V_0_y t + y_0, V_0_y = V_0 \sin\theta \\ y(t) =& \frac{-1}{2} gt^2 + V_0 \sin\theta t + y_0\\ x(t) =& 0 (g=0) + V_0_x t + x_0, V_0_x = V_0 \cos\theta \\ x(f) =& 0 (g=0) + V_0 \cos\theta t + x_0 \\ 0\ (end\ up\ on\ ground) = y_f = y(t_f) =& -\frac{1}{2}g t_f^2 + (v_0\sin\theta)t_f \\ R\ (want\ to\ travel\ R) = x_f = x(t_f) =& (v_0\cos\theta)t_f \\ \end{align}