Question

The speed of water flowing out from the small opening at a depth of \(h\) from the surface of water in a large tank is

• A. \(\sqrt{gh}\)
• B. \(\sqrt{4gh}\)
• C. \(gh\)
• D. \(2gh\)
• E. \(\sqrt{2gh}\)

Hint:

Torricelli's law shows that the speed of efflux from a small hole is the same as the speed a freely falling body would acquire after falling from rest through a distance \(h\).

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
This is Torricelli's law, derived from Bernoulli's principle.

Step 2: Detailed Explanation:
Consider a large tank with a small hole at a depth \(h\) below the free surface. Applying Bernoulli's equation at the free surface (point 1) and at the hole (point 2): \[ P_1 + \frac{1}{2}\rho v_1^2 + \rho g h_1 = P_2 + \frac{1}{2}\rho v_2^2 + \rho g h_2 \] Here, \(P_1 = P_2 =\) atmospheric pressure. The tank is large, so \(v_1 \approx 0\). Let \(h_1 - h_2 = h\). The equation simplifies to: \[ \rho g h = \frac{1}{2} \rho v_2^2 \] \[ v_2 = \sqrt{2gh} \]

Step 3: Final Answer:
The speed of efflux is \(\sqrt{2gh}\).