Question

A wide cylindrical vessel 50 cm in height is filled with water and rests on a table. Assuming the viscosity to be negligible, find at what height from the bottom of the vessel a small hole should be made for the water jet coming out of it to hit the surface of the table at the maximum horizontal distance from the vessel.

• A. 15 cm
• B. 35 cm
• C. 25 cm
• D. 10 cm

Answer 1

Answer: (C)

25 cm

Answer 2

Let $H$ be the total height of water (50 cm) and $h$ be the height of the hole from the bottom. The depth of the hole from the water surface is $d = H-h$. According to Torricelli's Law, the efflux velocity is $v = \sqrt{2gd} = \sqrt{2g(H-h)}$. The time of flight for the water jet to hit the table (falling a height $h$) is $t = \sqrt{2h/g}$. The horizontal range $x$ is given by $x = v \cdot t = \sqrt{2g(H-h)} \cdot \sqrt{2h/g} = 2\sqrt{h(H-h)}$. To maximize the horizontal range $x$, we need to maximize the term $h(H-h)$. This quadratic expression is maximized when $h = H/2$. Given $H = 50$ cm, the optimal height for the hole is $h = 50 \text{ cm} / 2 = 25 \text{ cm}$.