String Swing

The ball will follow the equations of projectile motion:

$\begin{aligned} x&=&-L\cos{\theta}+v_0\sin{\theta}\space t \\ y&=&L\sin{\theta}+v_0\cos{\theta}\space t-\frac{1}{2}gt^2 \end{aligned}$

We solve for $t$ when $y=0$ :

$t_+=\dfrac{v_0\cos{\theta}+\sqrt{v_0^2\cos^2{\theta}+2gL\sin{\theta}}}{g}$

We plug this result for the value of $t$ in the first equation to get the x-coordinate:

$x=-L\cos{\theta}+v_0\sin{\theta}\dfrac{v_0\cos{\theta}+\sqrt{v_0^2\cos^2{\theta}+2gL\sin{\theta}}}{g}$

To maximize $x(\theta)$ we should solve $\dfrac{dx}{d\theta}=0$ , which at first looks like it requires numerical treatment. So I decided to find the maximum using Geogebra (indulgence, please!). This is the plot and the numerical result I got:

First tried analytically, found an expression for x(θ) but setting its derivative to 0 got too messy (see pic below).

Then used Excel entering the formulas:

$x_0=-L\cos\theta$

$y_0=L\sin\theta$

$v_{x0}=v_0\sin\theta$

$v_{y0}=v_0\cos\theta$

$t=(v_{y0}+\sqrt{v_{y0}^2+2gy_0})/g$

$x(t)=x_0+v_{x0} t$

and tried values for θ. $θ=\boxed{62.917}$ gave the largest value for x.