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Understanding the speed of efflux plays a crucial role when analyzing fluid dynamics, particularly in scenarios involving the exit velocity of fluid through an orifice. Torricelli's theorem is instrumental in determining this speed, which is the velocity at which fluid leaves a hole in a container.

The formula derived from Torricelli's theorem is given by \( v = \sqrt{2gh} \), where \( v \) is the speed of efflux, \( g \) represents the acceleration due to gravity (approximately \( 9.81 m/s^2 \) on the surface of the Earth), and \( h \) is the height of the fluid above the hole. Think of this speed as the final velocity that a freely falling object would have after descending the height \( h \) under the influence of gravity.

Applying this knowledge to the exercise, with a water level 14.0 meters above the hole, we can use Torricelli's theorem to calculate the exit velocity or speed of efflux of the water.

Moving beyond the speed at which a fluid exits, we encounter the concept of volume discharge. This term refers to the quantity of fluid that flows out of an orifice per unit of time. To find the volume discharge, \( Q \), we multiply the speed of efflux, \( v \), by the cross-sectional area, \( A \), of the hole through which the fluid is discharged.

In mathematical terms, \( Q = A \times v \). The area \( A \) can be determined for circular holes with the formula \( A = \pi r^2 \), where \( r \) is the radius of the hole. The unit of volume discharge commonly used is cubic meters per second \( m^3/s \). Hence, for our exercise, once the speed of efflux is known, calculating the volume discharged per second becomes a straightforward multiplication.

Diving deeper, Bernoulli's theorem, also known as Bernoulli's principle, forms the foundation for understanding fluid behavior in various situations. It states that within a flowing fluid, there is a constant sum of potential energy, kinetic energy, and pressure energy.

The principle can be expressed as \( P + \frac{1}{2}\rho v^2 + \rho gh = \text{constant} \), where \( P \) stands for the fluid's pressure, \( \rho \) its density, \( v \) its velocity, and \( h \) the height relative to a reference point. Bernoulli's theorem illustrates that the sum of the fluid's pressure head, velocity head, and height head remains unchanged along a streamline.

Torricelli's theorem is actually a specific case of Bernoulli’s theorem. It's applied when a fluid exits into the air, which implies that the pressure at the exit point is atmospheric pressure and can be cancelled from the equation.

The acceleration due to gravity, denoted by \( g \), is the rate at which objects accelerate towards Earth without resistance, such as air resistance, being involved. Its standard value is \( 9.81 m/s^2 \). This acceleration is a key variable in equations for projectile motion, freefall, and, as we've seen, fluid dynamics with the Torricelli's theorem.

In our problem, the effect of gravity defines the speed of efflux as it is the force causing the water to accelerate as it exits the hole. It's important to note that while \( g \) is constant in many calculations, it can vary slightly depending on altitude and location on Earth, factors that are usually negligible in most high school or introductory college physics problems.