Question

Water and mercury are filled in two cylindrical vessels up to the same height. Both the vessels have a hole in the wall near the bottom. The velocity of water and mercury coming out of the holes are \(V_1\) and \(V_2\) respectively, then the relation between \(V_1\) and \(V_2\) is:

• A. \(V_1 = V_2\)
• B. \(V_1 = 13.6\,V_2\)
• C. \(V_1 = \dfrac{V_2}{13.6}\)
• D. \(V_1 = \sqrt{13.6}\,V_2\)

Hint:

Torricelli's law \(V=\sqrt{2gh}\) has no density in it, so equal heights give equal speeds.

Answer 1

The Correct Option is A

Step 1: Use Torricelli's law for efflux from a small hole. Applying Bernoulli's equation between the top surface and the hole gives the speed of the outflowing liquid: \[V=\sqrt{2gh},\] where \(h\) is the height of liquid above the hole.

Step 2: Notice that the density of the liquid cancels out in Bernoulli's equation. The efflux speed depends only on \(g\) and the height \(h\), not on the density.

Step 3: Both vessels are filled to the same height \(h\), so \[V_1=\sqrt{2gh}\quad\text{and}\quad V_2=\sqrt{2gh}.\] Step 4: Hence the two speeds are equal: \[V_1=V_2.\] \[\boxed{V_1=V_2}\]