Hole in One!

Let the velocity of the efflux be $v$ .

Using Torricelli's Law (a special case of Bernoulli's Principle), we observe that:

$v = \sqrt2(H - h)g$ ... (a)

Considering, the water stream to be parabolic, the time taken to hit the plane would be:

$t = \sqrt(\frac{2h}{g})$ ... (b)

We know that: $a = v \times t$

Therefore, putting (a) and (b), we get:

$a = \sqrt2(H - h)g \times \sqrt(\frac{2h}{g})$

$\implies$ $a = 2\sqrt(H - h)h$

In order to ensure the greatest distance, we would need to differentiate the expression within the square root w.r.t. $h$ .

$\frac{d}{dh}$ $(H - h)h$ = $H - 2h$ ... (c)

Differentiating again w.r.t. $h$ gives us $- 2$ , which is $< 0$ . Hence, we know that we are finding maxima.

Putting the expression (c) equal to zero:

$H - 2h = 0$

$\implies$ $h = \frac{H}{2} = \frac{12}{2} = \boxed{6}$