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: When water is flowing out of a hole at a certain height \( z \) from the bottom, it follows the principles of fluid dynamics. The velocity of the water emerging from the hole can be determined using Torricelli’s law:
   \[ v = \sqrt{2gz} \] where \( g \) is the acceleration due to gravity and \( z \) is the height from which the water is emerging.
2. Step 2: The horizontal range \( R \) of the water emerging from the hole depends on the velocity of the water and the height \( z \). The time of flight \( t \) for the water to reach the ground is given by:
   \[ t = \sqrt{\frac{2z}{g}} \] The horizontal range \( R \) can be found by multiplying the horizontal velocity \( v \) by the time of flight \( t \):
   \[ R = v \cdot t = \sqrt{2gz} \cdot \sqrt{\frac{2z}{g}} = 2z \]
3. Step 3: To maximize the range \( R \), we analyze \( R = 2z \). Differentiating \( R \) with respect to \( z \):
   \[ \frac{dR}{dz} = 2 \] Setting \( \frac{dR}{dz} = 0 \) does not apply here directly since the relationship is linear. However, the maximum range is achieved when the height \( z \) is proportional to the total height \( H \). From the geometry of the problem, the water has the most time to travel horizontally when \( z \) is one-third of \( H \).
4. Step 4: Therefore, the maximum range occurs when:
   \[ z = \frac{H}{3} \]

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

Answer 2

Maximum Range of Water Jet

Consider a cylindrical vessel of height H filled with water. A hole is made at a height z from the bottom.

1. Velocity of Efflux

According to Torricelli's theorem, the velocity of water emerging from the hole is:

v = √(2g(H - z))

where:

v is the velocity of efflux

g is the acceleration due to gravity

H is the total height of the water column

z is the height of the hole from the bottom

2. Time of Flight

The time taken for the water to fall a vertical distance z is given by:

t = √(2z/g)

where:

t is the time of flight

z is the vertical distance the water falls

g is the acceleration due to gravity

3. Horizontal Range

The horizontal range R is given by:

R = v × t = √(2g(H - z)) × √(2z/g)

R = √(4z(H - z)) = 2√(Hz - z²)

4. Maximizing the Range

To find the value of z for maximum range, we need to maximize the expression f(z) = Hz - z².

Taking the first derivative with respect to z and setting it to zero:

df(z)/dz = H - 2z = 0

2z = H

z = H/2

Taking the second derivative to confirm it's a maximum:

d²f(z)/dz² = -2

Since the second derivative is negative, the range is maximum at z = H/2.

Conclusion

The range of the emerging water through the hole will be maximum when the hole is made at a height of H/2 from the bottom.