A large tank filled with water to a height of h is said to...

A large tank filled with water to a height of $h$ is said to be emptied through a small hole at the bottom. The ratio of time taken for the level of water to fall down from $h$ to $h / 2$ and from $h / 2$ to zero is (a) $\sqrt{2}$ (b) $1 / \sqrt{2}$ (c) $\sqrt{2}-1$ (d) $\frac{1}{\sqrt{2}-1}$

A large tank filled with water to a height of $h$ is said to be emptied through a small hole at the bottom. The ratio of time taken for the level of water to fall down from $h$ to $h / 2$ and from $h / 2$ to zero is
(a) $\sqrt{2}$
(b) $1 / \sqrt{2}$
(c) $\sqrt{2}-1$
(d) $\frac{1}{\sqrt{2}-1}$

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Differential Equations in Fluid Dynamics

In scenarios involving the draining of a tank, differential equations are employed to link the rate of change of the fluid height with the velocity of outflow. This mathematical model encapsulates the continuous and dynamic nature of the water level as influenced by gravitational forces.

Ratio Analysis of Time Intervals

The analysis of the ratio of time intervals involves comparing the durations obtained for different ranges of water level decrease. This approach elucidates the relative dynamics over different stages of the fluid's fall, yielding insights into how changes in height affect the draining time.

Torricelli's Law

Torricelli's Law states that the speed at which fluid exits an orifice under the force of gravity is proportional to the square root of twice the gravitational acceleration times the height of the fluid above the opening. This principle is fundamental in fluid dynamics for relating the potential energy of the fluid to its kinetic energy as it leaves the tank.

Integration for Time Calculation

Integration is a crucial tool for solving the differential equations that describe the draining process. By integrating the rate of change of the fluid height over time, one can determine the time intervals required for the water level to drop between any two specified heights.

Transcript

Please give Ace some feedback