Elegant way to find maximum range of a projectile launched from a height

Question

A projectile is launched from a height of $h$ and the initial velocity
of $sqrt{2ga}$. Find the maximum range achieved by the projectile in terms of $g$,$a$, and $h$.

I can go about the traditional way to solve this.
First finding the time it takes to reach the ground and then writing the range in terms of $theta$, $g$,$a$, and $h$. Next, I can take the derivative to find the maximum.
But, that it is a tedious and messy process. Is there any way to do this without such messy manipulation of algebra?

Notwen · Accepted Answer

We put $y = -h$ and $x = R$ in equation of trajectory.
$$-h = Rtantheta - frac{gR^2}{2u^2}sec^2theta$$
$$-h = Rtantheta - frac{gR^2}{2u^2}(1 + tan^2theta)$$
Now we have a quadratic equation in $tantheta$. As $theta$ is real the determinant of quadratic equation should be always $geq 0$. The problem is quite easily solved when you do $b^2 -4ac > 0$ for it.

Elegant way to find maximum range of a projectile launched from a height

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