Problem 17. A vessel of cylindrical shape is filled with water. The height of the water layer is h. Water flows out of an opening (its dimensions can be neglected) located at height $z$ (Fig. 23). At what $z$ will the range of the water jet be the greatest?
## S o l u t i o n.

We introduce a rectangular coordinate system as shown in Figure 23. Let $v_{0}$ be the velocity of the water jet, $x$ be the distance from the $O x$ axis, and $y$ be the distance along the $O y$ axis. Then $v_{0}=\sqrt{2 g(h-z)}$, $x=v_{0} t, y=z-\frac{g t^{2}}{2}$. At the moment the water jet reaches the ground, $y=0$. From this, we have $t=\sqrt{\frac{2 z}{g}}$. Therefore, we have:

![](https://cdn.mathpix.com/cropped/2024_05_21_07b5a7f6f561b0ff634dg-22.jpg?height=368&width=443&top_left_y=1983&top_left_x=1275)

Fig. 23

![](https://cdn.mathpix.com/cropped/2024_05_21_07b5a7f6f561b0ff634dg-23.jpg?height=423&width=414&top_left_y=231&top_left_x=341)

Fig. 24

$$
\begin{gathered}
x=v_{0} \sqrt{\frac{2 z}{g}}=\sqrt{\frac{2 z}{g} \cdot 2 g(h-z)}= \\
=2 \sqrt{z(h-z)}
\end{gathered}
$$

$x$ reaches its maximum value when $z(h-z)$ is maximal. However, the sum of $z$ and $h-z$ is constant, so the product of non-negative factors reaches its maximum value when $z=h-z$. From this, we have

$$
z=\frac{h}{2}
$$