A tank is filled upto a height h with a liquid and is placed...

A tank is filled upto a height $h$ with a liquid and is placed on a platform of height $h$ from the ground. To get maximum range $x_{m}$, a small hole is punched at a distance of $y$ from the free surface of the liquid. Then
(a) $x_{m}=2 h$
(b) $x_{m}=1.5 h$
(c) $y=2 h$
(d) $y=0.75 h$

Key Concepts

Projectile Motion

Projectile motion studies the trajectory of an object launched into the air where horizontal and vertical motions are independent. The horizontal range of a projectile is determined by its initial speed and the height from which it is launched, along with gravitational acceleration. This concept is essential when analyzing the path and landing point of a fluid stream exiting a hole.

Torricelli's Law

Torricelli's Law states that the speed at which a liquid exits an opening under the force of gravity is given by the square root of (2g times the vertical distance between the free surface and the hole). This law applies energy conservation to fluid dynamics and is key to determining the exit velocity of the liquid.

Optimization in Fluid Trajectories

Optimization in this context involves determining the position of the hole (specifically, the vertical distance from the free surface) that maximizes the horizontal range of the exiting liquid stream. This is typically achieved by setting up an expression for the range in terms of this distance and finding its maximum via methods such as differentiation.

Energy Conservation in Fluid Flow

Energy conservation in fluid flow explains the conversion of gravitational potential energy of the fluid into kinetic energy as it exits the tank. This principle is used to derive the exit speed of the liquid and is fundamental in linking the height of the liquid column above the hole to the velocity of the stream.

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