Question

The water is filled upto height of $12\text{m}$ in a tank having vertical sidewalls. A hole is made in one of the walls at a depth ${}^{\prime }{h}^{\prime }$ below the waterlevel.The value of $h$ for which the emerging stream of waterstrikes the ground at the maximum range is $\text{(in m})$.

Answer 1

Let the asked situation be as shown in the below figure.

The range of water jet can be given as
$R=v\times t$, where $v$ is the velocity of jet and $t$ is the time taken by it to strike the ground.
Therefore, we have
$R=\sqrt{2gh}\times \sqrt{\frac{(12-h)\times 2}{g}}$
or, $R=\sqrt{4h(12-h)}$
For maximum range $R$, $\frac{dR}{dh}=0$
$\Rightarrow \frac{d\left(\sqrt{48h-4h^2}\right)}{dh}=\frac{(24-4h)}{\sqrt{48h-4h^2}}=0$
$\Rightarrow h=6\text{m}$