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

Derive the efflux speed by energy conservation for a small liquid element and see that mass (density) cancels.

Energy view. A small mass $\Delta m$ of liquid effectively falls from the top surface to the hole through height $h$. Its potential energy loss converts to kinetic energy: \[\Delta m\, g h=\tfrac{1}{2}\Delta m\, V^{2}.\] Cancel mass. The mass $\Delta m$ appears on both sides and cancels: \[gh=\tfrac{1}{2}V^{2}\;\Rightarrow\;V=\sqrt{2gh}.\] Apply to both liquids. This result has no density term. Water and mercury are filled to the same height $h$, so each comes out with the same speed: \[V_1=\sqrt{2gh}=V_2.\] Even though mercury is $13.6$ times denser than water, that density does not change the exit speed for the same head. \[\boxed{V_1=V_2}\]