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The speed of efflux is a crucial part of Torricelli's Law, which deals with how quickly a fluid exits a container through a small hole. Imagine a tank filled with water. Gravity pulls on the water and the pressure difference forces the water out through the hole. The speed of this efflux, or outflow, can be described mathematically.
The formula used here is:

• \( v = \sqrt{2gh} \), where \( v \) is the speed of efflux, \( g \) is the acceleration due to gravity, and \( h \) is the height of the water above the hole.

This equation shows that the speed of the water coming out depends on the height of the fluid. More height means more speed. This is because the higher the water level, the greater the pressure at the bottom, pushing fluid out more forcefully.
Understanding the speed of efflux is essential, as it helps predict how fast a tank will empty, based solely on the water level inside.

The height of the fluid in the tank is a dynamic factor that influences the flow rate through the hole. The initial height, \( H \), directly affects the pressure at the bottom of the tank.As the water flows out, the height changes, impacting the efflux speed. This process is the basis for determining the time it takes for water levels to decrease.
Here's how it works:

• Initially, with more water, the pressure is higher. Thus, water exits more quickly.
• As the water level drops, pressure decreases, slowing the outflow rate.

In the exercise, you have parts of the process where the height drops distinctively, first to \( \frac{H}{\eta} \) and then fully to zero. The variation in height is central to understanding how time intervals to empty the tank relate to one another, as shown by equating the given times \( T_1 \) and \( T_2 \).

Differential equations provide a valuable way to model how water flows out of the tank. They describe the relationship between changing variables—in this case, the water's height and time.The rate at which the height of the fluid changes with respect to time (\( \frac{dh}{dt} \)) is given by:

• \(-\frac{dh}{dt} = \frac{A}{a} \sqrt{2gh} \)

Here, \( A \) is the tank's cross-sectional area, and \( a \) is the hole's area. This equation tells us the rate of flow out of the tank depends inversely on the height of the water. As time progresses, the height decreases, affecting the rate of flow.
By solving differential equations, we can predict how long it will take for the tank to drain to specific heights, facilitating a deeper understanding of fluid mechanics.

Integration is a fundamental tool in physics used to calculate the results across continuous intervals. In the context of draining a tank, integration helps compute the total time it takes for the water level to change between certain heights.For the task, the integration process involves rearranging the equation to find \( dt \) and solving:

• \( dt = -\frac{a}{A \sqrt{2g}} \frac{1}{\sqrt{h}} dh \)

This differential equation is integrated between heights \( H \) to \( \frac{H}{\eta} \) for \( T_1 \) and from \( \frac{H}{\eta} \) to 0 for \( T_2 \). This gives:

• \( T_1 = \frac{2a}{A\sqrt{2g}}(\sqrt{H} - \sqrt{\frac{H}{\eta}}) \)
• \( T_2 = \frac{2a}{A\sqrt{2g}}\sqrt{\frac{H}{\eta}} \)

Integration is key to solving such problems because it allows us to accurately account for contributions over the decreasing fluid height rather than considering just discrete points.