A tank, which is open at the top, contains a liquid up to a...

A tank, which is open at the top, contains a liquid up to a height $H$. A small hole is made in the side of the tank at a distance $y$ below the liquid surface. The liquid emerging from the hole lands at a distance $x$ from the $\operatorname{tank}$ (A) If $y$ is increased from zero to $H, x$ will first increase and then decrease. (B) $x$ is maximum for $y=H / 2$. (C) The maximum value of $x$ is $H$. (D) The maximum value of $x$ will depend on the density of the liquid.

A tank, which is open at the top, contains a liquid up to a height $H$. A small hole is made in the side of the tank at a distance $y$ below the liquid surface. The liquid emerging from the hole lands at a distance $x$ from the $\operatorname{tank}$
(A) If $y$ is increased from zero to $H, x$ will first increase and then decrease.
(B) $x$ is maximum for $y=H / 2$.
(C) The maximum value of $x$ is $H$.
(D) The maximum value of $x$ will depend on the density of the liquid.

Key Concepts

Optimization in Fluid Mechanics

The problem involves finding the point along the tank’s side where the horizontal distance of the emerging stream is maximized. This requires setting up an equation for the range, which is a function of the height of the liquid above the hole, and determining its maximum value through techniques like differentiation. This concept illustrates how trade-offs between exit speed and time of flight can be optimized.

Independence from Liquid Density

In the derivation of the exit speed using Torricelli’s law, the density of the liquid cancels out, meaning the speed (and therefore the range) is independent of the liquid’s density. This emphasizes that for ideal fluid flow in such scenarios, the gravitational potential energy per unit mass is the driving factor, not the specific mass density of the fluid.

Torricelli’s Law

This law describes the speed of a fluid flowing out of an orifice under the influence of gravity. It states that the velocity of the liquid leaving the tank is equal to the square root of two times the gravitational acceleration multiplied by the height of the liquid above the hole. This concept is central to understanding how pressure head translates into kinetic energy as the fluid exits.

Projectile Motion

Once the liquid exits the tank, it follows a parabolic trajectory under the influence of gravity, similar to any projectile. The initial horizontal velocity, determined by Torricelli’s law, combined with the time of flight—which depends on the vertical distance the liquid falls—allows us to calculate the horizontal distance traveled by the liquid drop.

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