Question

Water is filled in a cylindrical vessel of height H. A hole is made at height z from the bottom, as shown in the figure. The value of z for which the range R of the emerging water through the hole will be maximum for:

• A. \( z = \frac{H}{4} \)
• B. \( z = \frac{H}{2} \)
• C. \( z = \frac{H}{8} \)
• D. \( z = \frac{H}{3} \)

Hint:

The range of the emerging water is maximized when the hole is located at the halfway point of the height of the water. This is because the time for the water to fall is balanced with the horizontal velocity.

Answer 1

The Correct Option is D

1. Step 1: Fluid dynamics govern water flow from a hole at height \( z \). Torricelli’s law gives the exit velocity:
   \[ v = \sqrt{2gz} \] where \( g \) is gravity and \( z \) is the height.
2. Step 2: The horizontal range \( R \) depends on velocity and height \( z \). Flight time \( t \) is:
   \[ t = \sqrt{\frac{2z}{g}} \] Range \( R \) is:
   \[ R = v \cdot t = \sqrt{2gz} \cdot \sqrt{\frac{2z}{g}} = 2z \]
3. Step 3: To maximize \( R \), consider \( R = 2z \). The derivative \( \frac{dR}{dz} = 2 \). Setting the derivative to zero doesn't apply directly here. The maximum range is when \( z \) is proportional to total height \( H \). The maximum horizontal travel time occurs when \( z \) is one-third of \( H \).
4. Step 4: Therefore, the maximum range is when:
   \[ z = \frac{H}{3} \]

Correct Answer: \( z = \frac{H}{3} \)